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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 38823. 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 38821, mpet3 38817, and with the conventional cpet 38819 and cpet2 38818, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38831 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.) |
Ref | Expression |
---|---|
mpet | ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpet3 38817 | . 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
2 | dfmembpart2 38751 | . 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) | |
3 | dfcomember3 38655 | . 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 206 ∧ wa 395 = wceq 1536 ∈ wcel 2105 ∅c0 4338 ∪ cuni 4911 / cqs 8742 ∼ ccoels 38162 CoElEqvRel wcoeleqvrel 38180 CoMembEr wcomember 38189 ElDisj weldisj 38197 MembPart wmembpart 38202 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1791 ax-4 1805 ax-5 1907 ax-6 1964 ax-7 2004 ax-8 2107 ax-9 2115 ax-10 2138 ax-11 2154 ax-12 2174 ax-ext 2705 ax-sep 5301 ax-nul 5311 ax-pr 5437 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1539 df-fal 1549 df-ex 1776 df-nf 1780 df-sb 2062 df-mo 2537 df-eu 2566 df-clab 2712 df-cleq 2726 df-clel 2813 df-nfc 2889 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3377 df-rab 3433 df-v 3479 df-dif 3965 df-un 3967 df-in 3969 df-ss 3979 df-nul 4339 df-if 4531 df-sn 4631 df-pr 4633 df-op 4637 df-uni 4912 df-br 5148 df-opab 5210 df-id 5582 df-eprel 5588 df-xp 5694 df-rel 5695 df-cnv 5696 df-co 5697 df-dm 5698 df-rn 5699 df-res 5700 df-ima 5701 df-ec 8745 df-qs 8749 df-coss 38392 df-coels 38393 df-refrel 38493 df-cnvrefrel 38508 df-symrel 38525 df-trrel 38555 df-eqvrel 38566 df-coeleqvrel 38568 df-dmqs 38620 df-erALTV 38645 df-comember 38647 df-funALTV 38663 df-disjALTV 38686 df-eldisj 38688 df-part 38747 df-membpart 38749 |
This theorem is referenced by: mpet2 38821 mainpart 38824 fences 38825 |
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