Theorem cpet2 37055

Description: The conventional form of the Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have called disjoint or partition what we call element disjoint or member partition, see also cpet 37056. Together with cpet 37056, mpet 37057 mpet2 37058, this is what we used to think of as the partition equivalence theorem (but cf. pet2 37068 with general 𝑅). (Contributed by Peter Mazsa, 30-Dec-2024.)

Assertion

Ref	Expression
cpet2	⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem cpet2

Step	Hyp	Ref	Expression
1		eldisjn0elb 36963	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴))
2		eqvrelqseqdisj3 37049	. . 3 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) → Disj (◡ E ↾ 𝐴))
3	2	petlem 37030	. 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (dom (◡ E ↾ 𝐴) / (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴))
4		eqvreldmqs2 36894	. 2 ⊢ (( EqvRel ≀ (◡ E ↾ 𝐴) ∧ (dom ≀ (◡ E ↾ 𝐴) / ≀ (◡ E ↾ 𝐴)) = 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
5	1, 3, 4	3bitri 296	1 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 396   = wceq 1540   ∈ wcel 2105  ∅c0 4267  ∪ cuni 4850   E cep 5512  ^◡ccnv 5606  dom cdm 5607   ↾ cres 5609   / cqs 8545   ≀ ccoss 36389   ∼ ccoels 36390   EqvRel weqvrel 36406   Disj wdisjALTV 36423   ElDisj weldisj 36425

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1912  ax-6 1970  ax-7 2010  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2153  ax-12 2170  ax-ext 2708  ax-sep 5238  ax-nul 5245  ax-pr 5367

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-nfc 2887  df-ne 2942  df-ral 3063  df-rex 3072  df-rmo 3350  df-rab 3405  df-v 3443  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4268  df-if 4472  df-sn 4572  df-pr 4574  df-op 4578  df-uni 4851  df-br 5088  df-opab 5150  df-id 5507  df-eprel 5513  df-xp 5613  df-rel 5614  df-cnv 5615  df-co 5616  df-dm 5617  df-rn 5618  df-res 5619  df-ima 5620  df-ec 8548  df-qs 8552  df-coss 36629  df-coels 36630  df-refrel 36730  df-cnvrefrel 36745  df-symrel 36762  df-trrel 36792  df-eqvrel 36803  df-funALTV 36900  df-disjALTV 36923  df-eldisj 36925

This theorem is referenced by:  cpet  37056