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Theorem compssiso 8822
 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 4545 . . . . 5
21ralrimivw 2810 . . . 4
3 compss.a . . . . 5
43fnmpt 5714 . . . 4
52, 4syl 17 . . 3
63compsscnv 8819 . . . . 5
76fneq1i 5680 . . . 4
85, 7sylibr 217 . . 3
9 dff1o4 5836 . . 3
105, 8, 9sylanbrc 677 . 2
11 elpwi 3951 . . . . . . . . 9
1211ad2antll 743 . . . . . . . 8
133isf34lem1 8820 . . . . . . . 8
1412, 13syldan 478 . . . . . . 7
15 elpwi 3951 . . . . . . . . 9
1615ad2antrl 742 . . . . . . . 8
173isf34lem1 8820 . . . . . . . 8
1816, 17syldan 478 . . . . . . 7
1914, 18psseq12d 3513 . . . . . 6
20 difss 3549 . . . . . . 7
21 pssdifcom1 3844 . . . . . . 7
2212, 20, 21sylancl 675 . . . . . 6
23 dfss4 3668 . . . . . . . 8
2416, 23sylib 201 . . . . . . 7
2524psseq1d 3511 . . . . . 6
2619, 22, 253bitrrd 288 . . . . 5
27 vex 3034 . . . . . 6
2827brrpss 6593 . . . . 5 []
29 fvex 5889 . . . . . 6
3029brrpss 6593 . . . . 5 []
3126, 28, 303bitr4g 296 . . . 4 [] []
32 relrpss 6591 . . . . 5 []
3332relbrcnv 5216 . . . 4 [] []
3431, 33syl6bbr 271 . . 3 [] []
3534ralrimivva 2814 . 2 [] []
36 df-isom 5598 . 2 [] [] [] []
3710, 35, 36sylanbrc 677 1 [] []
 Colors of variables: wff setvar class Syntax hints:   wi 4   wb 189   wa 376   wceq 1452   wcel 1904  wral 2756  cvv 3031   cdif 3387   wss 3390   wpss 3391  cpw 3942   class class class wbr 4395   cmpt 4454  ccnv 4838   wfn 5584  wf1o 5588  cfv 5589   wiso 5590   [] crpss 6589 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639 This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-rab 2765  df-v 3033  df-sbc 3256  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-pss 3406  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-isom 5598  df-rpss 6590 This theorem is referenced by:  isf34lem3  8823  isf34lem5  8826  isfin1-4  8835
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