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Theorem fiinbas 12225
 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 3117 . . . . . . . 8
2 eleq2 2203 . . . . . . . . . 10
3 sseq1 3120 . . . . . . . . . 10
42, 3anbi12d 464 . . . . . . . . 9
54rspcev 2789 . . . . . . . 8
61, 5mpanr2 434 . . . . . . 7
76ralrimiva 2505 . . . . . 6
87a1i 9 . . . . 5
98ralimdv 2500 . . . 4
109ralimdv 2500 . . 3
11 isbasis2g 12221 . . 3
1210, 11sylibrd 168 . 2
1312imp 123 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 103   wceq 1331   wcel 1480  wral 2416  wrex 2417   cin 3070   wss 3071  ctb 12218 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 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-v 2688  df-in 3077  df-ss 3084  df-pw 3512  df-uni 3737  df-bases 12219 This theorem is referenced by:  qtopbasss  12699
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