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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 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)) | |
4 | 3 | bicomi 223 | . . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ElDisj 𝐴) |
5 | cnvepresdmqs 37115 | . . 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 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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