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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 38206. 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 38204, mpet3 38200, and with the conventional cpet 38202 and cpet2 38201, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38214 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.) |
Ref | Expression |
---|---|
mpet | ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | mpet3 38200 | . 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
2 | dfmembpart2 38134 | . 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) | |
3 | dfcomember3 38038 | . 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 1533 ∈ wcel 2098 ∅c0 4315 ∪ cuni 4900 / cqs 8699 ∼ ccoels 37538 CoElEqvRel wcoeleqvrel 37556 CoMembEr wcomember 37565 ElDisj weldisj 37573 MembPart wmembpart 37578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2163 ax-ext 2695 ax-sep 5290 ax-nul 5297 ax-pr 5418 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-uni 4901 df-br 5140 df-opab 5202 df-id 5565 df-eprel 5571 df-xp 5673 df-rel 5674 df-cnv 5675 df-co 5676 df-dm 5677 df-rn 5678 df-res 5679 df-ima 5680 df-ec 8702 df-qs 8706 df-coss 37775 df-coels 37776 df-refrel 37876 df-cnvrefrel 37891 df-symrel 37908 df-trrel 37938 df-eqvrel 37949 df-coeleqvrel 37951 df-dmqs 38003 df-erALTV 38028 df-comember 38030 df-funALTV 38046 df-disjALTV 38069 df-eldisj 38071 df-part 38130 df-membpart 38132 |
This theorem is referenced by: mpet2 38204 mainpart 38207 fences 38208 |
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