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Theorem istpsOLD 19547
 Description: Express the predicate "is a topological space." (Contributed by NM, 18-Jul-2006.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
istpsOLD

Proof of Theorem istpsOLD
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 tpsexOLD 19546 . 2
2 simpr 461 . . . 4
3 uniexg 6596 . . . . 5
43adantr 465 . . . 4
52, 4eqeltrd 2545 . . 3
6 elex 3118 . . . 4
85, 7jca 532 . 2
9 df-topspOLD 19526 . . . 4
109eleq2i 2535 . . 3
11 eqeq1 2461 . . . . 5
1211anbi2d 703 . . . 4
13 eleq1 2529 . . . . 5
14 unieq 4259 . . . . . 6
1514eqeq2d 2471 . . . . 5
1613, 15anbi12d 710 . . . 4
1712, 16opelopabg 4774 . . 3
1810, 17syl5bb 257 . 2
191, 8, 18pm5.21nii 353 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wa 369   wceq 1395   wcel 1819  cvv 3109  cop 4038  cuni 4251  copab 4514  ctop 19520  ctpsOLD 19522 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-8 1821  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695  ax-un 6591 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-ral 2812  df-rex 2813  df-rab 2816  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-if 3945  df-sn 4033  df-pr 4035  df-op 4039  df-uni 4252  df-br 4457  df-opab 4516  df-xp 5014  df-rel 5015  df-topspOLD 19526 This theorem is referenced by:  istps2OLD  19548  retpsOLD  21396
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