Theorem cpet 38202

Description: The conventional form of Member Partition-Equivalence Theorem. In the conventional case there is no (general) disjoint and no (general) partition concept: mathematicians have been calling disjoint or partition what we call element disjoint or member partition, see also cpet2 38201. Cf. mpet 38203, mpet2 38204 and mpet3 38200 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet 38215 and pet2 38214 for Partition-Equivalence Theorem with general 𝑅. (Contributed by Peter Mazsa, 31-Dec-2024.)

Assertion

Ref	Expression
cpet	⊢ ( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Proof of Theorem cpet

Step	Hyp	Ref	Expression
1		dfmembpart2 38134	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2		cpet2 38201	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
3	1, 2	bitri 275	1 ⊢ ( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   = wceq 1533   ∈ wcel 2098  ∅c0 4315  ∪ cuni 4900   / cqs 8699   ∼ ccoels 37538   EqvRel weqvrel 37554   ElDisj weldisj 37573   MembPart wmembpart 37578

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2526  df-eu 2555  df-clab 2702  df-cleq 2716  df-clel 2802  df-nfc 2877  df-ne 2933  df-ral 3054  df-rex 3063  df-rmo 3368  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-uni 4901  df-br 5140  df-opab 5202  df-id 5565  df-eprel 5571  df-xp 5673  df-rel 5674  df-cnv 5675  df-co 5676  df-dm 5677  df-rn 5678  df-res 5679  df-ima 5680  df-ec 8702  df-qs 8706  df-coss 37775  df-coels 37776  df-refrel 37876  df-cnvrefrel 37891  df-symrel 37908  df-trrel 37938  df-eqvrel 37949  df-dmqs 38003  df-funALTV 38046  df-disjALTV 38069  df-eldisj 38071  df-part 38130  df-membpart 38132

This theorem is referenced by: (None)