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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 38769 | . 2 ⊢ ( MembPart 𝐴 ↔ (◡ E ↾ 𝐴) Part 𝐴) | |
| 2 | df-part 38767 | . 2 ⊢ ((◡ E ↾ 𝐴) Part 𝐴 ↔ ( Disj (◡ E ↾ 𝐴) ∧ (◡ E ↾ 𝐴) DomainQs 𝐴)) | |
| 3 | df-eldisj 38708 | . . . 4 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | |
| 4 | 3 | bicomi 224 | . . 3 ⊢ ( Disj (◡ E ↾ 𝐴) ↔ ElDisj 𝐴) | 
| 5 | cnvepresdmqs 38654 | . . 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)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: ¬ wn 3 ↔ wb 206 ∧ wa 395 ∈ wcel 2108 ∅c0 4333 E cep 5583 ◡ccnv 5684 ↾ cres 5687 DomainQs wdmqs 38206 Disj wdisjALTV 38216 ElDisj weldisj 38218 Part wpart 38221 MembPart wmembpart 38223 | 
| 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 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2708 ax-sep 5296 ax-nul 5306 ax-pr 5432 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3an 1089 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2540 df-eu 2569 df-clab 2715 df-cleq 2729 df-clel 2816 df-nfc 2892 df-ne 2941 df-ral 3062 df-rex 3071 df-rab 3437 df-v 3482 df-dif 3954 df-un 3956 df-in 3958 df-ss 3968 df-nul 4334 df-if 4526 df-sn 4627 df-pr 4629 df-op 4633 df-br 5144 df-opab 5206 df-eprel 5584 df-xp 5691 df-rel 5692 df-cnv 5693 df-dm 5695 df-rn 5696 df-res 5697 df-ima 5698 df-ec 8747 df-qs 8751 df-dmqs 38640 df-eldisj 38708 df-part 38767 df-membpart 38769 | 
| This theorem is referenced by: membpartlem19 38812 cpet 38839 mpet 38840 fences2 38846 | 
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