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Theorem sconntop 31809
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 31808 . 2 (𝐽 ∈ SConn → 𝐽 ∈ PConn)
2 pconntop 31806 . 2 (𝐽 ∈ PConn → 𝐽 ∈ Top)
31, 2syl 17 1 (𝐽 ∈ SConn → 𝐽 ∈ Top)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wcel 2107  Topctop 21105  PConncpconn 31800  SConncsconn 31801
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1839  ax-4 1853  ax-5 1953  ax-6 2021  ax-7 2055  ax-9 2116  ax-10 2135  ax-11 2150  ax-12 2163  ax-13 2334  ax-ext 2754
This theorem depends on definitions:  df-bi 199  df-an 387  df-or 837  df-3an 1073  df-tru 1605  df-ex 1824  df-nf 1828  df-sb 2012  df-clab 2764  df-cleq 2770  df-clel 2774  df-nfc 2921  df-ral 3095  df-rex 3096  df-rab 3099  df-v 3400  df-dif 3795  df-un 3797  df-in 3799  df-ss 3806  df-nul 4142  df-if 4308  df-sn 4399  df-pr 4401  df-op 4405  df-uni 4672  df-br 4887  df-iota 6099  df-fv 6143  df-ov 6925  df-pconn 31802  df-sconn 31803
This theorem is referenced by:  sconnpi1  31820  txsconn  31822  cvmlift3lem6  31905  cvmlift3lem7  31906  cvmlift3lem8  31907  cvmlift3lem9  31908
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