Theorem sconntop 35226

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 35225	. 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)
2		pconntop 35223	. 2 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
3	1, 2	syl 17	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2108  Topctop 22924  PConncpconn 35217  SConncsconn 35218

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1542  df-fal 1552  df-ex 1779  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3483  df-dif 3969  df-un 3971  df-ss 3983  df-nul 4343  df-if 4535  df-sn 4635  df-pr 4637  df-op 4641  df-uni 4916  df-br 5152  df-iota 6522  df-fv 6577  df-ov 7441  df-pconn 35219  df-sconn 35220

This theorem is referenced by:  sconnpi1  35237  txsconn  35239  cvmlift3lem6  35322  cvmlift3lem7  35323  cvmlift3lem8  35324  cvmlift3lem9  35325