The second-seeded Halep advanced with a 7-6 (3), 6-2 victory over former No. 1 Victoria Azarenka of Belarus.
Kerber will surrender the No 1 ranking on Monday to either Romanian Simona Halep, who must reach the semi-finals, or Karolina Pliskova.
Now she has the opportunity to take another step toward history in her meeting with Williams. "She's a tremendous champion, and I feel very, very humbled and I'm very excited to share the court with her again".
"I had a really tough time", Rybarikova said, "and right now, I'm here, and everything paid off".
He dropped the first two sets, then won the next two.
The 14th seed is coached by Conchita Martinez, the only Spanish player to have claimed the women's singles title at Wimbledon when she won in 1994.
In the 100th singles match of her Wimbledon career, Williams defeated Ostapenko 6-3, 7-5 in one hour and 13 minutes, to become the oldest semi-finalist here since Martina Navratilova. Williams used her service advantage to keep in control of the match, registering eight aces and being broken just once.
"Been working on that serve", she said.
Amusing how time works.
All Ostapenko could do afterward was rave about Williams' play. I did my best. "I think it's very hard for them to understand my position", she said.
Key stat: Rarely seen on today's circuit, Rybarikova's game - with her forward moving serve and delicately placed volleying - lends itself to the serve-and-volley approach. And to open things up even further, Jelena Ostapenko, the shock French Open victor, powered her way past fourth seed Elena Svitolina 6-3, 7-6 (6). Muguruza, though, showed very rarely to consider it as her best option.
"It's been something that I've been dragging back and forth for a while now", Djokovic said about his shoulder. "You need something extra" We all know Murray has all the shots, now it looks like he may have that little bit extra.
But Centre Court routinely takes two of its three daily matches from the men's draw, leaving some prominent female players feeling overlooked. "I earned the victory".
Key moment: This match was relatively even until early in the second set, when the wheels came off for Konjuh.
A strong clay court season led into the French Open, where she pulled off upset after upset with her powerful forehand being the key weapon.
If she should win the title, she could return to the top five, a full six and a half years since she was last there-and before contracting the Sjögren's Syndrome that nearly ended her career entirely. "I'd like to think that I can continue to rely on that as the matches continue".