Theorem mpet 39583

Description: Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 39586. 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 39584, mpet3 39580, and with the conventional cpet 39582 and cpet2 39581, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39594 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 39580	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfmembpart2 39503	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3		dfcomember3 39389	. 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
4	1, 2, 3	3bitr4i 306	1 ⊢ ( MembPart 𝐴 ↔ CoMembEr 𝐴)

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   = wceq 1570   ∈ wcel 2143  ∅c0 4287  ∪ cuni 4873   / cqs 8694   ∼ ccoels 38814   CoElEqvRel wcoeleqvrel 38832   CoMembEr wcomember 38843   ElDisj weldisj 38851   MembPart wmembpart 38856

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5258  ax-nul 5270  ax-pr 5406

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rmo 3369  df-rab 3417  df-v 3457  df-dif 3909  df-un 3911  df-in 3913  df-ss 3923  df-nul 4288  df-if 4489  df-sn 4591  df-pr 4593  df-op 4597  df-uni 4874  df-br 5111  df-opab 5175  df-id 5558  df-eprel 5563  df-xp 5669  df-rel 5670  df-cnv 5671  df-co 5672  df-dm 5673  df-rn 5674  df-res 5675  df-ima 5676  df-ec 8697  df-qs 8701  df-coss 39131  df-coels 39132  df-refrel 39222  df-cnvrefrel 39237  df-symrel 39254  df-trrel 39288  df-eqvrel 39299  df-coeleqvrel 39301  df-dmqs 39353  df-erALTV 39379  df-comember 39381  df-funALTV 39397  df-disjALTV 39420  df-eldisj 39422  df-part 39499  df-membpart 39501

This theorem is referenced by:  mpet2  39584  mainpart  39587  fences  39588