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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 38832. (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 38813 | . 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ 𝐴)) | |
2 | disjimxrn 38731 | . 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 1537 E cep 5588 ◡ccnv 5688 ↾ cres 5691 / cqs 8743 ⋉ cxrn 38161 EqvRel weqvrel 38179 Disj wdisjALTV 38196 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-10 2139 ax-11 2155 ax-12 2175 ax-ext 2706 ax-sep 5302 ax-nul 5312 ax-pr 5438 ax-un 7754 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1540 df-fal 1550 df-ex 1777 df-nf 1781 df-sb 2063 df-mo 2538 df-eu 2567 df-clab 2713 df-cleq 2727 df-clel 2814 df-nfc 2890 df-ne 2939 df-ral 3060 df-rex 3069 df-rmo 3378 df-rab 3434 df-v 3480 df-dif 3966 df-un 3968 df-in 3970 df-ss 3980 df-nul 4340 df-if 4532 df-sn 4632 df-pr 4634 df-op 4638 df-uni 4913 df-br 5149 df-opab 5211 df-mpt 5232 df-id 5583 df-eprel 5589 df-xp 5695 df-rel 5696 df-cnv 5697 df-co 5698 df-dm 5699 df-rn 5700 df-res 5701 df-ima 5702 df-iota 6516 df-fun 6565 df-fn 6566 df-f 6567 df-fo 6569 df-fv 6571 df-1st 8013 df-2nd 8014 df-ec 8746 df-qs 8750 df-xrn 38353 df-coss 38393 df-refrel 38494 df-cnvrefrel 38509 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 df-funALTV 38664 df-disjALTV 38687 df-eldisj 38689 |
This theorem is referenced by: pet2 38832 |
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