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Theorem fiinfi 36189
 Description: If two classes have the finite intersection property, then so does their intersection. (Contributed by Richard Penner, 1-Jan-2020.)
Hypotheses
Ref Expression
fiinfi.a
fiinfi.b
fiinfi.c
Assertion
Ref Expression
fiinfi
Distinct variable groups:   ,,   ,,   ,,   ,,

Proof of Theorem fiinfi
StepHypRef Expression
1 fiinfi.a . . . . . . 7
2 elinel1 3621 . . . . . . . . 9
3 elinel1 3621 . . . . . . . . . . 11
43imim1i 60 . . . . . . . . . 10
54ralimi2 2780 . . . . . . . . 9
62, 5imim12i 59 . . . . . . . 8
76ralimi2 2780 . . . . . . 7
81, 7syl 17 . . . . . 6
9 fiinfi.b . . . . . . 7
10 elinel2 3622 . . . . . . . . 9
11 elinel2 3622 . . . . . . . . . . 11
1211imim1i 60 . . . . . . . . . 10
1312ralimi2 2780 . . . . . . . . 9
1410, 13imim12i 59 . . . . . . . 8
1514ralimi2 2780 . . . . . . 7
169, 15syl 17 . . . . . 6
17 r19.26-2 2920 . . . . . 6
188, 16, 17sylanbrc 671 . . . . 5
19 elin 3619 . . . . . 6
20192ralbii 2822 . . . . 5
2118, 20sylibr 216 . . . 4
22 fiinfi.c . . . . . . 7
2322eleq2d 2516 . . . . . 6
2423ralbidv 2829 . . . . 5
2524ralbidv 2829 . . . 4
2621, 25mpbird 236 . . 3
2722raleqdv 2995 . . . 4
2827ralbidv 2829 . . 3
2926, 28mpbird 236 . 2
3022raleqdv 2995 . 2
3129, 30mpbird 236 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371   wceq 1446   wcel 1889  wral 2739   cin 3405 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433 This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ral 2744  df-v 3049  df-in 3413 This theorem is referenced by: (None)
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