Theorem sconntop 34674

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

Assertion

Ref	Expression
sconntop	⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top)

Proof of Theorem sconntop

Step	Hyp	Ref	Expression
1		sconnpconn 34673	. 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)
2		pconntop 34671	. 2 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
3	1, 2	syl 17	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2098  Topctop 22705  PConncpconn 34665  SConncsconn 34666

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3943  df-un 3945  df-in 3947  df-ss 3957  df-nul 4315  df-if 4521  df-sn 4621  df-pr 4623  df-op 4627  df-uni 4900  df-br 5139  df-iota 6485  df-fv 6541  df-ov 7404  df-pconn 34667  df-sconn 34668

This theorem is referenced by:  sconnpi1  34685  txsconn  34687  cvmlift3lem6  34770  cvmlift3lem7  34771  cvmlift3lem8  34772  cvmlift3lem9  34773