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Theorem dfmembpart2 37943
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 37941 . 2 ( MembPart 𝐴 ↔ ( E ↾ 𝐴) Part 𝐴)
2 df-part 37939 . 2 (( E ↾ 𝐴) Part 𝐴 ↔ ( Disj ( E ↾ 𝐴) ∧ ( E ↾ 𝐴) DomainQs 𝐴))
3 df-eldisj 37880 . . . 4 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
43bicomi 223 . . 3 ( Disj ( E ↾ 𝐴) ↔ ElDisj 𝐴)
5 cnvepresdmqs 37826 . . 3 (( E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴)
64, 5anbi12i 625 . 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 394  wcel 2104  c0 4321   E cep 5578  ccnv 5674  cres 5677   DomainQs wdmqs 37370   Disj wdisjALTV 37380   ElDisj weldisj 37382   Part wpart 37385   MembPart wmembpart 37387
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-eprel 5579  df-xp 5681  df-rel 5682  df-cnv 5683  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-ec 8707  df-qs 8711  df-dmqs 37812  df-eldisj 37880  df-part 37939  df-membpart 37941
This theorem is referenced by:  membpartlem19  37984  cpet  38011  mpet  38012  fences2  38018
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