Theorem scontop 30464

Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)

Assertion

Ref	Expression
scontop	⊢ (𝐽 ∈ SCon → 𝐽 ∈ Top)

Proof of Theorem scontop

Step	Hyp	Ref	Expression
1		sconpcon 30463	. 2 ⊢ (𝐽 ∈ SCon → 𝐽 ∈ PCon)
2		pcontop 30461	. 2 ⊢ (𝐽 ∈ PCon → 𝐽 ∈ Top)
3	1, 2	syl 17	1 ⊢ (𝐽 ∈ SCon → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 1977  Topctop 20517  PConcpcon 30455  SConcscon 30456

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-iota 5768  df-fv 5812  df-ov 6552  df-pcon 30457  df-scon 30458

This theorem is referenced by:  sconpi1  30475  txscon  30477  cvmlift3lem6  30560  cvmlift3lem7  30561  cvmlift3lem8  30562  cvmlift3lem9  30563