Theorem cpet 40021

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 40020. Cf. mpet 40022, mpet2 40023 and mpet3 40019 for unconventional forms of Member Partition-Equivalence Theorem. Cf. pet 40034 and pet2 40033 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 39942	. 2 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
2		cpet2 40020	. 2 ⊢ (( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴) ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))
3	1, 2	bitri 278	1 ⊢ ( MembPart 𝐴 ↔ ( EqvRel ∼ 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   ↔ wb 209   ∧ wa 401   = wceq 1569   ∈ wcel 2144  ∅c0 4277  ∪ cuni 4866   / cqs 8724   ∼ ccoels 39253   EqvRel weqvrel 39269   ElDisj weldisj 39290   MembPart wmembpart 39295

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 5391

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 3897  df-dif 4046  df-in 4090  df-un 4150  df-nul 4278  df-if 4482  df-sn 4585  df-pr 4587  df-op 4835  df-uni 4867  df-br 5103  df-opab 5167  df-id 5546  df-eprel 5551  df-xp 5657  df-rel 5658  df-cnv 5659  df-co 5660  df-dm 5661  df-rn 5662  df-res 5663  df-ima 5664  df-ec 8727  df-qs 8731  df-coss 39570  df-coels 39571  df-refrel 39661  df-cnvrefrel 39676  df-symrel 39693  df-trrel 39727  df-eqvrel 39738  df-dmqs 39792  df-funALTV 39836  df-disjALTV 39859  df-eldisj 39861  df-part 39938  df-membpart 39940

This theorem is used by: (None)