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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 38248 and mpet2 38249) 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 38217 | . 2 ⊢ (𝑅 Part 𝐴 → ≀ 𝑅 ErALTV 𝐴) | |
2 | mainer 38243 | . 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 37583 ErALTV werALTV 37609 CoMembEr wcomember 37611 Part wpart 37622 |
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 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-ec 8720 df-qs 8724 df-coss 37820 df-coels 37821 df-refrel 37921 df-cnvrefrel 37936 df-symrel 37953 df-trrel 37983 df-eqvrel 37994 df-coeleqvrel 37996 df-dmqs 38048 df-erALTV 38073 df-comember 38075 df-funALTV 38091 df-disjALTV 38114 df-eldisj 38116 df-part 38175 |
This theorem is referenced by: mainpart 38252 |
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