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Mirrors > Home > MPE Home > Th. List > Mathboxes > partimcomember | Structured version Visualization version GIF version |
Description: Partition with general 𝑅 (in addition to the member partition cf. mpet 38782 and mpet2 38783) implies equivalent comembers. (Contributed by Peter Mazsa, 23-Sep-2021.) (Revised by Peter Mazsa, 22-Dec-2024.) |
Ref | Expression |
---|---|
partimcomember | ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | partim 38751 | . 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | |
2 | mainer 38777 | . 2 ⊢ ( ≀ 𝑅 ErALTV 𝐴 → CoMembEr 𝐴) | |
3 | 1, 2 | syl 17 | 1 ⊢ (𝑅 Part 𝐴 → CoMembEr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ≀ ccoss 38122 ErALTV werALTV 38148 CoMembEr wcomember 38150 Part wpart 38161 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1963 ax-7 2003 ax-8 2106 ax-9 2114 ax-10 2137 ax-11 2153 ax-12 2173 ax-ext 2704 ax-sep 5300 ax-nul 5307 ax-pr 5430 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 847 df-3an 1087 df-tru 1538 df-fal 1548 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2536 df-eu 2565 df-clab 2711 df-cleq 2725 df-clel 2812 df-nfc 2888 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3376 df-rab 3433 df-v 3479 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4915 df-br 5150 df-opab 5212 df-id 5576 df-eprel 5582 df-xp 5689 df-rel 5690 df-cnv 5691 df-co 5692 df-dm 5693 df-rn 5694 df-res 5695 df-ima 5696 df-ec 8740 df-qs 8744 df-coss 38354 df-coels 38355 df-refrel 38455 df-cnvrefrel 38470 df-symrel 38487 df-trrel 38517 df-eqvrel 38528 df-coeleqvrel 38530 df-dmqs 38582 df-erALTV 38607 df-comember 38609 df-funALTV 38625 df-disjALTV 38648 df-eldisj 38650 df-part 38709 |
This theorem is referenced by: mainpart 38786 |
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