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Theorem compsscnv 8289
 Description: Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.)
Hypothesis
Ref Expression
compss.a
Assertion
Ref Expression
compsscnv
Distinct variable group:   ,
Allowed substitution hint:   ()

Proof of Theorem compsscnv
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 cnvopab 5309 . 2
2 compss.a . . . 4
3 difeq2 3448 . . . . 5
43cbvmptv 4331 . . . 4
5 df-mpt 4299 . . . 4
62, 4, 53eqtri 2467 . . 3
76cnveqi 5082 . 2
8 df-mpt 4299 . . 3
9 compsscnvlem 8288 . . . . 5
10 compsscnvlem 8288 . . . . 5
119, 10impbii 182 . . . 4
1211opabbii 4303 . . 3
138, 2, 123eqtr4i 2473 . 2
141, 7, 133eqtr4i 2473 1
 Colors of variables: wff set class Syntax hints:   wa 360   wceq 1654   wcel 1728   cdif 3306  cpw 3828  copab 4296   cmpt 4297  ccnv 4912 This theorem is referenced by:  compssiso  8292  isf34lem3  8293  compss  8294  isf34lem5  8296 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1628  ax-9 1669  ax-8 1690  ax-14 1732  ax-6 1747  ax-7 1752  ax-11 1764  ax-12 1954  ax-ext 2424  ax-sep 4361  ax-nul 4369  ax-pr 4438 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1661  df-eu 2292  df-mo 2293  df-clab 2430  df-cleq 2436  df-clel 2439  df-nfc 2568  df-ne 2608  df-ral 2717  df-rex 2718  df-rab 2721  df-v 2967  df-dif 3312  df-un 3314  df-in 3316  df-ss 3323  df-nul 3617  df-if 3768  df-pw 3830  df-sn 3849  df-pr 3850  df-op 3852  df-br 4244  df-opab 4298  df-mpt 4299  df-xp 4919  df-rel 4920  df-cnv 4921
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