Theorem dfmembpart2 39412

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 39410	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
2		df-part 39408	. 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴))
3		df-eldisj 39331	. . . 4 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
4	3	bicomi 227	. . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ElDisj 𝐴)
5		cnvepresdmqs 39277	. . 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 2149  ∅c0 4294   E cep 5561  ^◡ccnv 5661   ↾ cres 5664   DomainQs wdmqs 38746   Disj wdisjALTV 38758   ElDisj weldisj 38760   Part wpart 38763   MembPart wmembpart 38765

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-10 2182  ax-11 2198  ax-12 2219  ax-ext 2741  ax-sep 5261  ax-nul 5271  ax-pr 5405

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-nf 1811  df-sb 2098  df-mo 2573  df-eu 2603  df-clab 2748  df-cleq 2761  df-clel 2844  df-nfc 2918  df-ne 2965  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5114  df-opab 5178  df-eprel 5562  df-xp 5668  df-rel 5669  df-cnv 5670  df-dm 5672  df-rn 5673  df-res 5674  df-ima 5675  df-ec 8696  df-qs 8700  df-dmqs 39262  df-eldisj 39331  df-part 39408  df-membpart 39410

This theorem is referenced by:  membpartlem19  39453  cpet  39491  mpet  39492  fences2  39498