Theorem sconntop 35810

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 35809	. 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)
2		pconntop 35807	. 2 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
3	1, 2	syl 18	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top)

This proof depends on syntax axioms:   → wi 4   ∈ wcel 2145  Topctop 23121  PConncpconn 35801  SConncsconn 35802

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2732

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-iota 6489  df-fv 6541  df-ov 7417  df-pconn 35803  df-sconn 35804

This theorem is used by:  sconnpi1  35821  txsconn  35823  cvmlift3lem6  35906  cvmlift3lem7  35907  cvmlift3lem8  35908  cvmlift3lem9  35909