Theorem sconntop 35215

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 35214	. 2 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ PConn)
2		pconntop 35212	. 2 ⊢ (𝐽 ∈ PConn → 𝐽 ∈ Top)
3	1, 2	syl 17	1 ⊢ (𝐽 ∈ SConn → 𝐽 ∈ Top)

Syntax hints:   → wi 4   ∈ wcel 2109  Topctop 22786  PConncpconn 35206  SConncsconn 35207

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-ss 3933  df-nul 4299  df-if 4491  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-iota 6466  df-fv 6521  df-ov 7392  df-pconn 35208  df-sconn 35209

This theorem is referenced by:  sconnpi1  35226  txsconn  35228  cvmlift3lem6  35311  cvmlift3lem7  35312  cvmlift3lem8  35313  cvmlift3lem9  35314