Theorem mpet 39709

Description: Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 39712. 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 39710, mpet3 39706, and with the conventional cpet 39708 and cpet2 39707, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39720 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 39706	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfmembpart2 39629	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3		dfcomember3 39515	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
4	1, 2, 3	3bitr4i 306	1 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2145  ∅c0 4282  ∪ cuni 4870   / cqs 8699   ∼ ccoels 38940   CoElEqvRel wcoeleqvrel 38958   CoMembEr wcomember 38969   ElDisj weldisj 38977   MembPart wmembpart 38982

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-10 2178  ax-11 2194  ax-12 2215  ax-ext 2734  ax-sep 5255  ax-nul 5267  ax-pr 5402

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-mo 2566  df-eu 2596  df-clab 2741  df-cleq 2754  df-clel 2837  df-nfc 2911  df-ne 2958  df-ral 3079  df-rex 3089  df-rmo 3367  df-rab 3415  df-v 3455  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4283  df-if 4486  df-sn 4588  df-pr 4590  df-op 4594  df-uni 4871  df-br 5108  df-opab 5172  df-id 5554  df-eprel 5559  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-res 5671  df-ima 5672  df-ec 8702  df-qs 8706  df-coss 39257  df-coels 39258  df-refrel 39348  df-cnvrefrel 39363  df-symrel 39380  df-trrel 39414  df-eqvrel 39425  df-coeleqvrel 39427  df-dmqs 39479  df-erALTV 39505  df-comember 39507  df-funALTV 39523  df-disjALTV 39546  df-eldisj 39548  df-part 39625  df-membpart 39627

This theorem is used by:  mpet2  39710  mainpart  39713  fences  39714