Theorem dfmembpart2 38752

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 38750	. 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴)
2		df-part 38748	. 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴))
3		df-eldisj 38689	. . . 4 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴))
4	3	bicomi 224	. . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ElDisj 𝐴)
5		cnvepresdmqs 38635	. . 3 ⊢ ((◡ E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
6	4, 5	anbi12i 628	. 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴) ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
7	1, 2, 6	3bitri 297	1 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))

Syntax hints:  ¬ wn 3   ↔ wb 206   ∧ wa 395   ∈ wcel 2106  ∅c0 4339   E cep 5588  ^◡ccnv 5688   ↾ cres 5691   DomainQs wdmqs 38186   Disj wdisjALTV 38196   ElDisj weldisj 38198   Part wpart 38201   MembPart wmembpart 38203

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750  df-dmqs 38621  df-eldisj 38689  df-part 38748  df-membpart 38750

This theorem is referenced by:  membpartlem19  38793  cpet  38820  mpet  38821  fences2  38827