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Theorem isbasisg 12402
 Description: Express the predicate "the set is a basis for a topology". (Contributed by NM, 17-Jul-2006.)
Assertion
Ref Expression
isbasisg
Distinct variable group:   ,,
Allowed substitution hints:   (,)

Proof of Theorem isbasisg
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 ineq1 3301 . . . . . 6
21unieqd 3783 . . . . 5
32sseq2d 3158 . . . 4
43raleqbi1dv 2660 . . 3
54raleqbi1dv 2660 . 2
6 df-bases 12401 . 2
75, 6elab2g 2859 1
 Colors of variables: wff set class Syntax hints:   wi 4   wb 104   wceq 1335   wcel 2128  wral 2435   cin 3101   wss 3102  cpw 3543  cuni 3772  ctb 12400 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ral 2440  df-rex 2441  df-v 2714  df-in 3108  df-ss 3115  df-uni 3773  df-bases 12401 This theorem is referenced by:  isbasis2g  12403  basis1  12405  baspartn  12408
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