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Theorem sconntop 30971
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
StepHypRef Expression
1 sconnpconn 30970 . 2 (𝐽 ∈ SConn → 𝐽 ∈ PConn)
2 pconntop 30968 . 2 (𝐽 ∈ PConn → 𝐽 ∈ Top)
31, 2syl 17 1 (𝐽 ∈ SConn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 1987  Topctop 20638  PConncpconn 30962  SConncsconn 30963
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-iota 5820  df-fv 5865  df-ov 6618  df-pconn 30964  df-sconn 30965
This theorem is referenced by:  sconnpi1  30982  txsconn  30984  cvmlift3lem6  31067  cvmlift3lem7  31068  cvmlift3lem8  31069  cvmlift3lem9  31070
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