Theorem mpet 38787

Description: Member Partition-Equivalence Theorem in almost its shortest possible form, cf. the 0-ary version mpets 38790. 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 38788, mpet3 38784, and with the conventional cpet 38786 and cpet2 38785, this is what we used to think of as the partition equivalence theorem (but cf. pet2 38798 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 38784	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
2		dfmembpart2 38718	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
3		dfcomember3 38622	. 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 1537   ∈ wcel 2108  ∅c0 4352  ∪ cuni 4931   / cqs 8756   ∼ ccoels 38128   CoElEqvRel wcoeleqvrel 38146   CoMembEr wcomember 38155   ElDisj weldisj 38163   MembPart wmembpart 38168

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2158  ax-12 2178  ax-ext 2711  ax-sep 5317  ax-nul 5324  ax-pr 5447

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 847  df-3an 1089  df-tru 1540  df-fal 1550  df-ex 1778  df-nf 1782  df-sb 2065  df-mo 2543  df-eu 2572  df-clab 2718  df-cleq 2732  df-clel 2819  df-nfc 2895  df-ne 2947  df-ral 3068  df-rex 3077  df-rmo 3388  df-rab 3444  df-v 3490  df-dif 3979  df-un 3981  df-in 3983  df-ss 3993  df-nul 4353  df-if 4549  df-sn 4649  df-pr 4651  df-op 4655  df-uni 4932  df-br 5167  df-opab 5229  df-id 5593  df-eprel 5599  df-xp 5701  df-rel 5702  df-cnv 5703  df-co 5704  df-dm 5705  df-rn 5706  df-res 5707  df-ima 5708  df-ec 8759  df-qs 8763  df-coss 38359  df-coels 38360  df-refrel 38460  df-cnvrefrel 38475  df-symrel 38492  df-trrel 38522  df-eqvrel 38533  df-coeleqvrel 38535  df-dmqs 38587  df-erALTV 38612  df-comember 38614  df-funALTV 38630  df-disjALTV 38653  df-eldisj 38655  df-part 38714  df-membpart 38716

This theorem is referenced by:  mpet2  38788  mainpart  38791  fences  38792