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Mirrors > Home > MPE Home > Th. List > Mathboxes > dfmembpart2 | Structured version Visualization version GIF version |
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.) |
Ref | Expression |
---|---|
dfmembpart2 | ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | df-membpart 37730 | . 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴) | |
2 | df-part 37728 | . 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴)) | |
3 | df-eldisj 37669 | . . . 4 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | |
4 | 3 | bicomi 223 | . . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ElDisj 𝐴) |
5 | cnvepresdmqs 37615 | . . 3 ⊢ ((◡ E ↾ 𝐴) DomainQs 𝐴 ↔ ¬ ∅ ∈ 𝐴) | |
6 | 4, 5 | anbi12i 627 | . 2 ⊢ (( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴) ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) |
7 | 1, 2, 6 | 3bitri 296 | 1 ⊢ ( MembPart 𝐴 ↔ ( ElDisj 𝐴 ∧ ¬ ∅ ∈ 𝐴)) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 ↔ wb 205 ∧ wa 396 ∈ wcel 2106 ∅c0 4322 E cep 5579 ◡ccnv 5675 ↾ cres 5678 DomainQs wdmqs 37159 Disj wdisjALTV 37169 ElDisj weldisj 37171 Part wpart 37174 MembPart wmembpart 37176 |
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 2703 ax-sep 5299 ax-nul 5306 ax-pr 5427 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3951 df-un 3953 df-in 3955 df-ss 3965 df-nul 4323 df-if 4529 df-sn 4629 df-pr 4631 df-op 4635 df-br 5149 df-opab 5211 df-eprel 5580 df-xp 5682 df-rel 5683 df-cnv 5684 df-dm 5686 df-rn 5687 df-res 5688 df-ima 5689 df-ec 8707 df-qs 8711 df-dmqs 37601 df-eldisj 37669 df-part 37728 df-membpart 37730 |
This theorem is referenced by: membpartlem19 37773 cpet 37800 mpet 37801 fences2 37807 |
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