Theorem mpet3 39640

Description: Member Partition-Equivalence Theorem. Together with mpet 39643 mpet2 39644, mostly in its conventional cpet 39642 and cpet2 39641 form, this is what we used to think of as the partition equivalence theorem (but cf. pet2 39654 with general 𝑅). (Contributed by Peter Mazsa, 4-May-2018.) (Revised by Peter Mazsa, 26-Sep-2021.)

Assertion

Ref	Expression
mpet3	⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem mpet3

Step	Hyp	Ref	Expression
1		eldisjn0elb 39535	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
2		eqvrelqseqdisj3 39635	. . 3 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) → Disj (◡ E ↾ 𝐴))
3	2	petlem 39605	. 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
4		eqvreldmqs 39450	. 2 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
5	1, 3, 4	3bitri 300	1 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1570   ∈ wcel 2146  ∅c0 4289  ∪ cuni 4877   E cep 5565  ^◡ccnv 5665  dom cdm 5666   ↾ cres 5668   / cqs 8702   ≀ ccoss 38873   ∼ ccoels 38874   EqvRel weqvrel 38890   CoElEqvRel wcoeleqvrel 38892   Disj wdisjALTV 38909   ElDisj weldisj 38911

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 2148  ax-9 2156  ax-10 2179  ax-11 2195  ax-12 2216  ax-ext 2738  ax-sep 5262  ax-nul 5274  ax-pr 5409

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 2570  df-eu 2600  df-clab 2745  df-cleq 2758  df-clel 2841  df-nfc 2915  df-ne 2962  df-ral 3083  df-rex 3093  df-rmo 3372  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-uni 4878  df-br 5115  df-opab 5179  df-id 5561  df-eprel 5566  df-xp 5672  df-rel 5673  df-cnv 5674  df-co 5675  df-dm 5676  df-rn 5677  df-res 5678  df-ima 5679  df-ec 8705  df-qs 8709  df-coss 39191  df-coels 39192  df-refrel 39282  df-cnvrefrel 39297  df-symrel 39314  df-trrel 39348  df-eqvrel 39359  df-coeleqvrel 39361  df-funALTV 39457  df-disjALTV 39480  df-eldisj 39482

This theorem is used by:  mpet  39643