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Theorem compssiso 8254
 Description: Complementation is an antiautomorphism on power set lattices. (Contributed by Stefan O'Rear, 4-Nov-2014.) (Proof shortened by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a
Assertion
Ref Expression
compssiso [] []
Distinct variable groups:   ,   ,
Allowed substitution hint:   ()

Proof of Theorem compssiso
Dummy variables are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 difexg 4351 . . . . 5
21ralrimivw 2790 . . . 4
3 compss.a . . . . 5
43fnmpt 5571 . . . 4
52, 4syl 16 . . 3
63compsscnv 8251 . . . . 5
76fneq1i 5539 . . . 4
85, 7sylibr 204 . . 3
9 dff1o4 5682 . . 3
105, 8, 9sylanbrc 646 . 2
11 elpwi 3807 . . . . . . . . 9
1211ad2antll 710 . . . . . . . 8
133isf34lem1 8252 . . . . . . . 8
1412, 13syldan 457 . . . . . . 7
15 elpwi 3807 . . . . . . . . 9
1615ad2antrl 709 . . . . . . . 8
173isf34lem1 8252 . . . . . . . 8
1816, 17syldan 457 . . . . . . 7
1914, 18psseq12d 3441 . . . . . 6
20 difss 3474 . . . . . . 7
21 pssdifcom1 3713 . . . . . . 7
2212, 20, 21sylancl 644 . . . . . 6
23 dfss4 3575 . . . . . . . 8
2416, 23sylib 189 . . . . . . 7
2524psseq1d 3439 . . . . . 6
2619, 22, 253bitrrd 272 . . . . 5
27 vex 2959 . . . . . 6
2827brrpss 6525 . . . . 5 []
29 fvex 5742 . . . . . 6
3029brrpss 6525 . . . . 5 []
3126, 28, 303bitr4g 280 . . . 4 [] []
32 relrpss 6523 . . . . 5 []
3332relbrcnv 5245 . . . 4 [] []
3431, 33syl6bbr 255 . . 3 [] []
3534ralrimivva 2798 . 2 [] []
36 df-isom 5463 . 2 [] [] [] []
3710, 35, 36sylanbrc 646 1 [] []
 Colors of variables: wff set class Syntax hints:   wi 4   wb 177   wa 359   wceq 1652   wcel 1725  wral 2705  cvv 2956   cdif 3317   wss 3320   wpss 3321  cpw 3799   class class class wbr 4212   cmpt 4266  ccnv 4877   wfn 5449  wf1o 5453  cfv 5454   wiso 5455   [] crpss 6521 This theorem is referenced by:  isf34lem3  8255  isf34lem5  8258  isfin1-4  8267 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-sep 4330  ax-nul 4338  ax-pr 4403 This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-eu 2285  df-mo 2286  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-ral 2710  df-rex 2711  df-rab 2714  df-v 2958  df-sbc 3162  df-dif 3323  df-un 3325  df-in 3327  df-ss 3334  df-pss 3336  df-nul 3629  df-if 3740  df-pw 3801  df-sn 3820  df-pr 3821  df-op 3823  df-uni 4016  df-br 4213  df-opab 4267  df-mpt 4268  df-id 4498  df-xp 4884  df-rel 4885  df-cnv 4886  df-co 4887  df-dm 4888  df-rn 4889  df-iota 5418  df-fun 5456  df-fn 5457  df-f 5458  df-f1 5459  df-fo 5460  df-f1o 5461  df-fv 5462  df-isom 5463  df-rpss 6522
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