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Mirrors > Home > MPE Home > Th. List > compsscnv | Unicode version |
Description: Complementation on a power set lattice is an involution. (Contributed by Mario Carneiro, 17-May-2015.) |
Ref | Expression |
---|---|
compss.a |
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Ref | Expression |
---|---|
compsscnv |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | cnvopab 5241 |
. 2
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2 | compss.a |
. . . 4
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3 | difeq2 3427 |
. . . . 5
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4 | 3 | cbvmptv 4268 |
. . . 4
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5 | df-mpt 4236 |
. . . 4
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6 | 2, 4, 5 | 3eqtri 2436 |
. . 3
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7 | 6 | cnveqi 5014 |
. 2
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8 | df-mpt 4236 |
. . 3
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9 | compsscnvlem 8214 |
. . . . 5
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10 | compsscnvlem 8214 |
. . . . 5
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11 | 9, 10 | impbii 181 |
. . . 4
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12 | 11 | opabbii 4240 |
. . 3
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13 | 8, 2, 12 | 3eqtr4i 2442 |
. 2
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14 | 1, 7, 13 | 3eqtr4i 2442 |
1
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Colors of variables: wff set class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem is referenced by: compssiso 8218 isf34lem3 8219 compss 8220 isf34lem5 8222 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1552 ax-5 1563 ax-17 1623 ax-9 1662 ax-8 1683 ax-14 1725 ax-6 1740 ax-7 1745 ax-11 1757 ax-12 1946 ax-ext 2393 ax-sep 4298 ax-nul 4306 ax-pr 4371 |
This theorem depends on definitions: df-bi 178 df-or 360 df-an 361 df-3an 938 df-tru 1325 df-ex 1548 df-nf 1551 df-sb 1656 df-eu 2266 df-mo 2267 df-clab 2399 df-cleq 2405 df-clel 2408 df-nfc 2537 df-ne 2577 df-ral 2679 df-rex 2680 df-rab 2683 df-v 2926 df-dif 3291 df-un 3293 df-in 3295 df-ss 3302 df-nul 3597 df-if 3708 df-pw 3769 df-sn 3788 df-pr 3789 df-op 3791 df-br 4181 df-opab 4235 df-mpt 4236 df-xp 4851 df-rel 4852 df-cnv 4853 |
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