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Theorem dfmembpart2 37232
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
StepHypRef Expression
1 df-membpart 37230 . 2 ( MembPart 𝐴 ↔ ( E ↾ 𝐴) Part 𝐴)
2 df-part 37228 . 2 (( E ↾ 𝐴) Part 𝐴 ↔ ( Disj ( E ↾ 𝐴) ∧ ( E ↾ 𝐴) DomainQs 𝐴))
3 df-eldisj 37169 . . . 4 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
43bicomi 223 . . 3 ( Disj ( E ↾ 𝐴) ↔ ElDisj 𝐴)
5 cnvepresdmqs 37115 . . 3 (( E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
64, 5anbi12i 627 . 2 (( Disj ( E ↾ 𝐴) ∧ ( E ↾ 𝐴) DomainQs 𝐴) ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
71, 2, 63bitri 296 1 ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wb 205  wa 396  wcel 2106  c0 4282   E cep 5536  ccnv 5632  cres 5635   DomainQs wdmqs 36658   Disj wdisjALTV 36668   ElDisj weldisj 36670   Part wpart 36673   MembPart wmembpart 36675
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5256  ax-nul 5263  ax-pr 5384
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-mo 2538  df-eu 2567  df-clab 2714  df-cleq 2728  df-clel 2814  df-nfc 2889  df-ne 2944  df-ral 3065  df-rex 3074  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-opab 5168  df-eprel 5537  df-xp 5639  df-rel 5640  df-cnv 5641  df-dm 5643  df-rn 5644  df-res 5645  df-ima 5646  df-ec 8650  df-qs 8654  df-dmqs 37101  df-eldisj 37169  df-part 37228  df-membpart 37230
This theorem is referenced by:  membpartlem19  37273  cpet  37300  mpet  37301  fences2  37307
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