Theorem mpet 39979

Description: Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 39982. 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 39980, mpet3 39976, and with the conventional cpet 39978 and cpet2 39977, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39990 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 39976	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfmembpart2 39899	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3		dfcomember3 39785	. 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 1569   ∈ wcel 2144  ∅c0 4278  ∪ cuni 4866   / cqs 8720   ∼ ccoels 39210   CoElEqvRel wcoeleqvrel 39228   CoMembEr wcomember 39239   ElDisj weldisj 39247   MembPart wmembpart 39252

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-10 2177  ax-11 2193  ax-12 2212  ax-ext 2732  ax-sep 5248  ax-nul 5259  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-nf 1816  df-sb 2099  df-mo 2564  df-clab 2739  df-cleq 2752  df-clel 2835  df-nfc 2909  df-ne 2956  df-ral 3077  df-rex 3087  df-rmo 3365  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5545  df-eprel 5550  df-xp 5656  df-rel 5657  df-cnv 5658  df-co 5659  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663  df-ec 8723  df-qs 8727  df-coss 39527  df-coels 39528  df-refrel 39618  df-cnvrefrel 39633  df-symrel 39650  df-trrel 39684  df-eqvrel 39695  df-coeleqvrel 39697  df-dmqs 39749  df-erALTV 39775  df-comember 39777  df-funALTV 39793  df-disjALTV 39816  df-eldisj 39818  df-part 39895  df-membpart 39897

This theorem is used by:  mpet2  39980  mainpart  39983  fences  39984