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Theorem fisub 25657
 Description: If a set has the finite intersection property, its subsets have also this property. (Contributed by FL, 27-Apr-2008.)
Hypotheses
Ref Expression
fisub.1
fisub.2
Assertion
Ref Expression
fisub
Distinct variable groups:   ,,   ,   ,,
Allowed substitution hints:   ()   (,)

Proof of Theorem fisub
StepHypRef Expression
1 0ex 4166 . . . 4
2 eqeq1 2302 . . . . . 6
323anbi3d 1258 . . . . 5
43exbidv 1616 . . . 4
5 fisub.2 . . . 4
61, 4, 5elab2 2930 . . 3
7 sstr 3200 . . . . . . . . 9
823anbi3d 1258 . . . . . . . . . . . . . 14
98exbidv 1616 . . . . . . . . . . . . 13
10 fisub.1 . . . . . . . . . . . . 13
111, 9, 10elab2 2930 . . . . . . . . . . . 12
1211biimpri 197 . . . . . . . . . . 11
131219.23bi 1814 . . . . . . . . . 10
14133exp 1150 . . . . . . . . 9
157, 14syl 15 . . . . . . . 8
1615expcom 424 . . . . . . 7
1716com4l 78 . . . . . 6
18173imp 1145 . . . . 5
1918exlimiv 1624 . . . 4
2019com12 27 . . 3
216, 20syl5bi 208 . 2
2221con3d 125 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 358   w3a 934  wex 1531   wceq 1632   wcel 1696  cab 2282   wss 3165  c0 3468  cint 3878  cfn 6879 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-in 3172  df-ss 3179  df-nul 3469
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