Theorem dfmembpart2 39542

Description: Alternate definition of the conventional membership case of partition. Partition 𝐴 of 𝑋, [Halmos] p. 28: "A partition of 𝑋 is a disjoint collection 𝐴 of non-empty subsets of 𝑋 whose union is 𝑋", or Definition 35, [Suppes] p. 83., cf. https://oeis.org/A000110 . (Contributed by Peter Mazsa, 14-Aug-2021.)

Assertion

Ref	Expression
dfmembpart2	⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

Proof of Theorem dfmembpart2

Step	Hyp	Ref	Expression
1		df-membpart 39540	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
2		df-part 39538	. 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴))
3		df-eldisj 39461	. . . 4 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
4	3	bicomi 227	. . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ElDisj 𝐴)
5		cnvepresdmqs 39407	. . 3 ⊢ ((◡ E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
6	4, 5	anbi12i 639	. 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴) ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
7	1, 2, 6	3bitri 300	1 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 209   ∧ wa 400   ∈ wcel 2143  ∅c0 4286   E cep 5560  ^◡ccnv 5660   ↾ cres 5663   DomainQs wdmqs 38876   Disj wdisjALTV 38888   ElDisj weldisj 38890   Part wpart 38893   MembPart wmembpart 38895

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-10 2176  ax-11 2192  ax-12 2213  ax-ext 2735  ax-sep 5257  ax-nul 5269  ax-pr 5404

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-nf 1814  df-sb 2097  df-mo 2567  df-eu 2597  df-clab 2742  df-cleq 2755  df-clel 2838  df-nfc 2912  df-ne 2959  df-ral 3080  df-rex 3090  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-in 3912  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-opab 5174  df-eprel 5561  df-xp 5667  df-rel 5668  df-cnv 5669  df-dm 5671  df-rn 5672  df-res 5673  df-ima 5674  df-ec 8692  df-qs 8696  df-dmqs 39392  df-eldisj 39461  df-part 39538  df-membpart 39540

This theorem is referenced by:  membpartlem19  39583  cpet  39621  mpet  39622  fences2  39628