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Mirrors > Home > MPE Home > Th. List > Mathboxes > eldisjim | Structured version Visualization version GIF version |
Description: If the elements of 𝐴 are disjoint, then it has equivalent coelements (former prter1 38383). Special case of disjim 38285. (Contributed by Rodolfo Medina, 13-Oct-2010.) (Revised by Mario Carneiro, 12-Aug-2015) (Revised by Peter Mazsa, 8-Feb-2018.) ( Revised by Peter Mazsa, 23-Sep-2021.) |
Ref | Expression |
---|---|
eldisjim | ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | disjim 38285 | . 2 ⊢ ( Disj (◡ E ↾ 𝐴) → EqvRel ≀ (◡ E ↾ 𝐴)) | |
2 | df-eldisj 38211 | . 2 ⊢ ( ElDisj 𝐴 ↔ Disj (◡ E ↾ 𝐴)) | |
3 | df-coeleqvrel 38091 | . 2 ⊢ ( CoElEqvRel 𝐴 ↔ EqvRel ≀ (◡ E ↾ 𝐴)) | |
4 | 1, 2, 3 | 3imtr4i 291 | 1 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 E cep 5585 ◡ccnv 5681 ↾ cres 5684 ≀ ccoss 37681 EqvRel weqvrel 37698 CoElEqvRel wcoeleqvrel 37700 Disj wdisjALTV 37715 ElDisj weldisj 37717 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-10 2129 ax-11 2146 ax-12 2166 ax-ext 2699 ax-sep 5303 ax-nul 5310 ax-pr 5433 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 846 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2529 df-eu 2558 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ral 3059 df-rex 3068 df-rab 3431 df-v 3475 df-dif 3952 df-un 3954 df-in 3956 df-ss 3966 df-nul 4327 df-if 4533 df-sn 4633 df-pr 4635 df-op 4639 df-br 5153 df-opab 5215 df-id 5580 df-xp 5688 df-rel 5689 df-cnv 5690 df-co 5691 df-dm 5692 df-rn 5693 df-res 5694 df-coss 37915 df-refrel 38016 df-cnvrefrel 38031 df-symrel 38048 df-trrel 38078 df-eqvrel 38089 df-coeleqvrel 38091 df-disjALTV 38209 df-eldisj 38211 |
This theorem is referenced by: mainer 38338 mainer2 38350 |
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