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Theorem isufil 20272
 Description: The property of being an ultrafilter. (Contributed by Jeff Hankins, 30-Nov-2009.) (Revised by Mario Carneiro, 29-Jul-2015.)
Assertion
Ref Expression
isufil
Distinct variable groups:   ,   ,

Proof of Theorem isufil
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-ufil 20270 . 2
2 pweq 4019 . . . 4
4 eleq2 2540 . . . . 5
54adantl 466 . . . 4
6 difeq1 3620 . . . . 5
7 eleq12 2543 . . . . 5
86, 7sylan 471 . . . 4
95, 8orbi12d 709 . . 3
103, 9raleqbidv 3077 . 2
11 fveq2 5872 . 2
12 fvex 5882 . 2
13 elfvdm 5898 . 2
141, 10, 11, 12, 13elmptrab2 20197 1
 Colors of variables: wff setvar class Syntax hints:   wb 184   wo 368   wa 369   wceq 1379   wcel 1767  wral 2817  cvv 3118   cdif 3478  cpw 4016   cdm 5005  cfv 5594  cfil 20214  cufil 20268 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4574  ax-nul 4582  ax-pow 4631  ax-pr 4692 This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2822  df-rex 2823  df-rab 2826  df-v 3120  df-sbc 3337  df-csb 3441  df-dif 3484  df-un 3486  df-in 3488  df-ss 3495  df-nul 3791  df-if 3946  df-pw 4018  df-sn 4034  df-pr 4036  df-op 4040  df-uni 4252  df-br 4454  df-opab 4512  df-mpt 4513  df-id 4801  df-xp 5011  df-rel 5012  df-cnv 5013  df-co 5014  df-dm 5015  df-rn 5016  df-res 5017  df-ima 5018  df-iota 5557  df-fun 5596  df-fv 5602  df-ufil 20270 This theorem is referenced by:  ufilfil  20273  ufilss  20274  isufil2  20277  trufil  20279  fixufil  20291  fin1aufil  20301
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