Theorem mpet 38831

Description: Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 38834. 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 38832, mpet3 38828, and with the conventional cpet 38830 and cpet2 38829, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38842 with general 𝑅). (Contributed by Peter Mazsa, 24-Sep-2021.)

Assertion

Ref	Expression
mpet	⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)

Proof of Theorem mpet

Step	Hyp	Ref	Expression
1		mpet3 38828	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfmembpart2 38762	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3		dfcomember3 38666	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
4	1, 2, 3	3bitr4i 303	1 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   = wceq 1540   ∈ wcel 2109  ∅c0 4296  ∪ cuni 4871   / cqs 8670   ∼ ccoels 38170   CoElEqvRel wcoeleqvrel 38188   CoMembEr wcomember 38197   ElDisj weldisj 38205   MembPart wmembpart 38210

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 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5251  ax-nul 5261  ax-pr 5387

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-ral 3045  df-rex 3054  df-rmo 3354  df-rab 3406  df-v 3449  df-dif 3917  df-un 3919  df-in 3921  df-ss 3931  df-nul 4297  df-if 4489  df-sn 4590  df-pr 4592  df-op 4596  df-uni 4872  df-br 5108  df-opab 5170  df-id 5533  df-eprel 5538  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-ec 8673  df-qs 8677  df-coss 38402  df-coels 38403  df-refrel 38503  df-cnvrefrel 38518  df-symrel 38535  df-trrel 38565  df-eqvrel 38576  df-coeleqvrel 38578  df-dmqs 38630  df-erALTV 38656  df-comember 38658  df-funALTV 38674  df-disjALTV 38697  df-eldisj 38699  df-part 38758  df-membpart 38760

This theorem is referenced by:  mpet2  38832  mainpart  38835  fences  38836