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Mirrors > Home > MPE Home > Th. List > Mathboxes > mpet | Structured version Visualization version GIF version |
Description: Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 38308. Member partition and comember equivalence relation are the same (or: each element of 𝐴 have equivalent comembers if and only if 𝐴 is a member partition). Together with mpet2 38306, mpet3 38302, and with the conventional cpet 38304 and cpet2 38303, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38316 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.) |
Ref | Expression |
---|---|
mpet | ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpet3 38302 | . 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
2 | dfmembpart2 38236 | . 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) | |
3 | dfcomember3 38140 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
4 | 1, 2, 3 | 3bitr4i 303 | 1 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∅c0 4318 ∪ cuni 4903 / cqs 8717 ∼ ccoels 37643 CoElEqvRel wcoeleqvrel 37661 CoMembEr wcomember 37670 ElDisj weldisj 37678 MembPart wmembpart 37683 |
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 2167 ax-ext 2699 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 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2937 df-ral 3058 df-rex 3067 df-rmo 3372 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 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 37877 df-coels 37878 df-refrel 37978 df-cnvrefrel 37993 df-symrel 38010 df-trrel 38040 df-eqvrel 38051 df-coeleqvrel 38053 df-dmqs 38105 df-erALTV 38130 df-comember 38132 df-funALTV 38148 df-disjALTV 38171 df-eldisj 38173 df-part 38232 df-membpart 38234 |
This theorem is referenced by: mpet2 38306 mainpart 38309 fences 38310 |
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