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Theorem pcontop 27251
Description: A simply connected space is a topology. (Contributed by Mario Carneiro, 11-Feb-2015.)
Assertion
Ref Expression
pcontop  |-  ( J  e. PCon  ->  J  e.  Top )

Proof of Theorem pcontop
Dummy variables  f  x  y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2451 . . 3  |-  U. J  =  U. J
21ispcon 27249 . 2  |-  ( J  e. PCon 
<->  ( J  e.  Top  /\ 
A. x  e.  U. J A. y  e.  U. J E. f  e.  ( II  Cn  J ) ( ( f ` 
0 )  =  x  /\  ( f ` 
1 )  =  y ) ) )
32simplbi 460 1  |-  ( J  e. PCon  ->  J  e.  Top )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    /\ wa 369    = wceq 1370    e. wcel 1758   A.wral 2795   E.wrex 2796   U.cuni 4192   ` cfv 5519  (class class class)co 6193   0cc0 9386   1c1 9387   Topctop 18623    Cn ccn 18953   IIcii 20576  PConcpcon 27245
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1952  ax-ext 2430
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2437  df-cleq 2443  df-clel 2446  df-nfc 2601  df-ral 2800  df-rex 2801  df-rab 2804  df-v 3073  df-dif 3432  df-un 3434  df-in 3436  df-ss 3443  df-nul 3739  df-if 3893  df-sn 3979  df-pr 3981  df-op 3985  df-uni 4193  df-br 4394  df-iota 5482  df-fv 5527  df-ov 6196  df-pcon 27247
This theorem is referenced by:  scontop  27254  pconcon  27257  txpcon  27258  ptpcon  27259  qtoppcon  27262  pconpi1  27263  sconpi1  27265  cvxscon  27269
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