Theorem sconntop 35720

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 35719	. 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)
2		pconntop 35717	. 2 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
3	1, 2	syl 18	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2143  Topctop 23050  PConncpconn 35711  SConncsconn 35712

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4873  df-br 5110  df-iota 6492  df-fv 6544  df-ov 7413  df-pconn 35713  df-sconn 35714

This theorem is referenced by:  sconnpi1  35731  txsconn  35733  cvmlift3lem6  35816  cvmlift3lem7  35817  cvmlift3lem8  35818  cvmlift3lem9  35819