Theorem dfmembpart2 37944

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 37942	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
2		df-part 37940	. 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴))
3		df-eldisj 37881	. . . 4 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
4	3	bicomi 223	. . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ElDisj 𝐴)
5		cnvepresdmqs 37827	. . 3 ⊢ ((◡ E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
6	4, 5	anbi12i 626	. 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴) ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
7	1, 2, 6	3bitri 296	1 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 205   ∧ wa 395   ∈ wcel 2105  ∅c0 4323   E cep 5580  ^◡ccnv 5676   ↾ cres 5679   DomainQs wdmqs 37371   Disj wdisjALTV 37381   ElDisj weldisj 37383   Part wpart 37386   MembPart wmembpart 37388

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 2702  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1781  df-nf 1785  df-sb 2067  df-mo 2533  df-eu 2562  df-clab 2709  df-cleq 2723  df-clel 2809  df-nfc 2884  df-ne 2940  df-ral 3061  df-rex 3070  df-rab 3432  df-v 3475  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-eprel 5581  df-xp 5683  df-rel 5684  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690  df-ec 8708  df-qs 8712  df-dmqs 37813  df-eldisj 37881  df-part 37940  df-membpart 37942

This theorem is referenced by:  membpartlem19  37985  cpet  38012  mpet  38013  fences2  38019