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Theorem isbasis2g 17005
 Description: Express the predicate " is a basis for a topology." (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
isbasis2g
Distinct variable group:   ,,,,
Allowed substitution hints:   (,,,)

Proof of Theorem isbasis2g
StepHypRef Expression
1 isbasisg 17004 . 2
2 dfss3 3330 . . . 4
3 elin 3522 . . . . . . . . . 10
4 df-pw 3793 . . . . . . . . . . . 12
54abeq2i 2542 . . . . . . . . . . 11
65anbi2i 676 . . . . . . . . . 10
73, 6bitri 241 . . . . . . . . 9
87anbi2i 676 . . . . . . . 8
9 an12 773 . . . . . . . 8
108, 9bitri 241 . . . . . . 7
1110exbii 1592 . . . . . 6
12 eluni 4010 . . . . . 6
13 df-rex 2703 . . . . . 6
1411, 12, 133bitr4i 269 . . . . 5
1514ralbii 2721 . . . 4
162, 15bitri 241 . . 3
17162ralbii 2723 . 2
181, 17syl6bb 253 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359  wex 1550   wcel 1725  wral 2697  wrex 2698   cin 3311   wss 3312  cpw 3791  cuni 4007  ctb 16954 This theorem is referenced by:  isbasis3g  17006  basis2  17008  fiinbas  17009  tgclb  17027  topbas  17029  restbas  17214  txbas  17591  blbas  18452 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 2416 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ral 2702  df-rex 2703  df-v 2950  df-in 3319  df-ss 3326  df-pw 3793  df-uni 4008  df-bases 16957
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