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Theorem topbas 17037
 Description: A topology is its own basis. (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
topbas

Proof of Theorem topbas
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 inopn 16972 . . . . . . . 8
213expb 1154 . . . . . . 7
32adantr 452 . . . . . 6
4 simpr 448 . . . . . . 7
5 ssid 3367 . . . . . . 7
64, 5jctir 525 . . . . . 6
7 eleq2 2497 . . . . . . . 8
8 sseq1 3369 . . . . . . . 8
97, 8anbi12d 692 . . . . . . 7
109rspcev 3052 . . . . . 6
113, 6, 10syl2anc 643 . . . . 5
1211exp31 588 . . . 4
1312ralrimdv 2795 . . 3
1413ralrimivv 2797 . 2
15 isbasis2g 17013 . 2
1614, 15mpbird 224 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 359   wceq 1652   wcel 1725  wral 2705  wrex 2706   cin 3319   wss 3320  ctop 16958  ctb 16962 This theorem is referenced by:  resttop  17224  dis1stc  17562  txtop  17601  onpsstopbas  26180 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ral 2710  df-rex 2711  df-v 2958  df-in 3327  df-ss 3334  df-pw 3801  df-uni 4016  df-top 16963  df-bases 16965
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