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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 38843. 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 38841, mpet3 38837, and with the conventional cpet 38839 and cpet2 38838, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38851 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.) |
| Ref | Expression |
|---|---|
| mpet | ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴) |
| Step | Hyp | Ref | Expression |
|---|---|---|---|
| 1 | mpet3 38837 | . 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
| 2 | dfmembpart2 38771 | . 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) | |
| 3 | dfcomember3 38675 | . 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 1540 ∈ wcel 2108 ∅c0 4333 ∪ cuni 4907 / cqs 8744 ∼ ccoels 38183 CoElEqvRel wcoeleqvrel 38201 CoMembEr wcomember 38210 ElDisj weldisj 38218 MembPart wmembpart 38223 |
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 |
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3380 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-uni 4908 df-br 5144 df-opab 5206 df-id 5578 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-co 5694 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 df-coss 38412 df-coels 38413 df-refrel 38513 df-cnvrefrel 38528 df-symrel 38545 df-trrel 38575 df-eqvrel 38586 df-coeleqvrel 38588 df-dmqs 38640 df-erALTV 38665 df-comember 38667 df-funALTV 38683 df-disjALTV 38706 df-eldisj 38708 df-part 38767 df-membpart 38769 |
| This theorem is referenced by: mpet2 38841 mainpart 38844 fences 38845 |
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