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Theorem topbas 19600
 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 19534 . . . . . . . 8
213expb 1197 . . . . . . 7
32adantr 465 . . . . . 6
4 simpr 461 . . . . . . 7
5 ssid 3518 . . . . . . 7
64, 5jctir 538 . . . . . 6
7 eleq2 2530 . . . . . . . 8
8 sseq1 3520 . . . . . . . 8
97, 8anbi12d 710 . . . . . . 7
109rspcev 3210 . . . . . 6
113, 6, 10syl2anc 661 . . . . 5
1211exp31 604 . . . 4
1312ralrimdv 2873 . . 3
1413ralrimivv 2877 . 2
15 isbasis2g 19575 . 2
1614, 15mpbird 232 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 369   wceq 1395   wcel 1819  wral 2807  wrex 2808   cin 3470   wss 3471  ctop 19520  ctb 19524 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-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578 This theorem depends on definitions:  df-bi 185  df-an 371  df-3an 975  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ral 2812  df-rex 2813  df-v 3111  df-in 3478  df-ss 3485  df-pw 4017  df-uni 4252  df-top 19525  df-bases 19527 This theorem is referenced by:  resttop  19787  dis1stc  20125  txtop  20195  onpsstopbas  30057
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