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Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelqseqdisj5 | Structured version Visualization version GIF version |
Description: Lemma for the Partition-Equivalence Theorem pet2 38259. (Contributed by Peter Mazsa, 15-Jul-2020.) (Revised by Peter Mazsa, 22-Sep-2021.) |
Ref | Expression |
---|---|
eqvrelqseqdisj5 | ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ⋉ (◡ E ↾ 𝐴))) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelqseqdisj3 38240 | . 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ 𝐴)) | |
2 | disjimxrn 38158 | . 2 ⊢ ( Disj (◡ E ↾ 𝐴) → Disj (𝑆 ⋉ (◡ E ↾ 𝐴))) | |
3 | 1, 2 | syl 17 | 1 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (𝑆 ⋉ (◡ E ↾ 𝐴))) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 = wceq 1534 E cep 5575 ◡ccnv 5671 ↾ cres 5674 / cqs 8717 ⋉ cxrn 37582 EqvRel weqvrel 37600 Disj wdisjALTV 37617 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pr 5423 ax-un 7734 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-ral 3057 df-rex 3066 df-rmo 3371 df-rab 3428 df-v 3471 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-eprel 5576 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-fo 6548 df-fv 6550 df-1st 7987 df-2nd 7988 df-ec 8720 df-qs 8724 df-xrn 37780 df-coss 37820 df-refrel 37921 df-cnvrefrel 37936 df-symrel 37953 df-trrel 37983 df-eqvrel 37994 df-funALTV 38091 df-disjALTV 38114 df-eldisj 38116 |
This theorem is referenced by: pet2 38259 |
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