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Theorem fiinbas 17019
 Description: If a set is closed under finite intersection, then it is a basis for a topology. (Contributed by Jeff Madsen, 2-Sep-2009.)
Assertion
Ref Expression
fiinbas
Distinct variable groups:   ,,   ,,

Proof of Theorem fiinbas
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 ssid 3369 . . . . . . . 8
2 eleq2 2499 . . . . . . . . . 10
3 sseq1 3371 . . . . . . . . . 10
42, 3anbi12d 693 . . . . . . . . 9
54rspcev 3054 . . . . . . . 8
61, 5mpanr2 667 . . . . . . 7
76ralrimiva 2791 . . . . . 6
87a1i 11 . . . . 5
98ralimdv 2787 . . . 4
109ralimdv 2787 . . 3
11 isbasis2g 17015 . . 3
1210, 11sylibrd 227 . 2
1312imp 420 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 360   wceq 1653   wcel 1726  wral 2707  wrex 2708   cin 3321   wss 3322  ctb 16964 This theorem is referenced by:  fibas  17044  qtopbaslem  18794  ontopbas  26180 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ral 2712  df-rex 2713  df-v 2960  df-in 3329  df-ss 3336  df-pw 3803  df-uni 4018  df-bases 16967
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