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Mirrors > Home > MPE Home > Th. List > disj | Structured version Visualization version Unicode version |
Description: Two ways of saying that two classes are disjoint (have no members in common). (Contributed by NM, 17-Feb-2004.) |
Ref | Expression |
---|---|
disj |
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Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-in 3422 |
. . . 4
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2 | 1 | eqeq1i 2466 |
. . 3
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3 | abeq1 2571 |
. . 3
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4 | imnan 428 |
. . . . 5
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5 | noel 3746 |
. . . . . 6
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6 | 5 | nbn 353 |
. . . . 5
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7 | 4, 6 | bitr2i 258 |
. . . 4
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8 | 7 | albii 1701 |
. . 3
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9 | 2, 3, 8 | 3bitri 279 |
. 2
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10 | df-ral 2753 |
. 2
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11 | 9, 10 | bitr4i 260 |
1
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Colors of variables: wff setvar class |
Syntax hints: ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() ![]() |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1679 ax-4 1692 ax-5 1768 ax-6 1815 ax-7 1861 ax-10 1925 ax-11 1930 ax-12 1943 ax-13 2101 ax-ext 2441 |
This theorem depends on definitions: df-bi 190 df-an 377 df-tru 1457 df-ex 1674 df-nf 1678 df-sb 1808 df-clab 2448 df-cleq 2454 df-clel 2457 df-nfc 2591 df-ral 2753 df-v 3058 df-dif 3418 df-in 3422 df-nul 3743 |
This theorem is referenced by: disjr 3817 disj1 3818 disjne 3821 otiunsndisj 4720 onxpdisj 5560 f0rn0 5790 onint 6648 zfreg 8135 kmlem4 8608 fin23lem30 8797 fin23lem31 8798 isf32lem3 8810 fpwwe2 9093 renfdisj 9719 fvinim0ffz 12054 metdsge 21914 metdsgeOLD 21929 spthispth 25351 2spotdisj 25837 2spotiundisj 25838 2spotmdisj 25844 subfacp1lem1 29950 dfpo2 30443 dvmptfprodlem 37856 stoweidlem26 37923 stoweidlem59 37957 iundjiunlem 38334 otiunsndisjX 39045 |
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