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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 38176. (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 38157 | . 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ 𝐴)) | |
2 | disjimxrn 38075 | . 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 1533 E cep 5569 ◡ccnv 5665 ↾ cres 5668 / cqs 8697 ⋉ cxrn 37498 EqvRel weqvrel 37516 Disj wdisjALTV 37533 |
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 2163 ax-ext 2695 ax-sep 5289 ax-nul 5296 ax-pr 5417 ax-un 7718 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-nf 1778 df-sb 2060 df-mo 2526 df-eu 2555 df-clab 2702 df-cleq 2716 df-clel 2802 df-nfc 2877 df-ne 2933 df-ral 3054 df-rex 3063 df-rmo 3368 df-rab 3425 df-v 3468 df-dif 3943 df-un 3945 df-in 3947 df-ss 3957 df-nul 4315 df-if 4521 df-sn 4621 df-pr 4623 df-op 4627 df-uni 4900 df-br 5139 df-opab 5201 df-mpt 5222 df-id 5564 df-eprel 5570 df-xp 5672 df-rel 5673 df-cnv 5674 df-co 5675 df-dm 5676 df-rn 5677 df-res 5678 df-ima 5679 df-iota 6485 df-fun 6535 df-fn 6536 df-f 6537 df-fo 6539 df-fv 6541 df-1st 7968 df-2nd 7969 df-ec 8700 df-qs 8704 df-xrn 37697 df-coss 37737 df-refrel 37838 df-cnvrefrel 37853 df-symrel 37870 df-trrel 37900 df-eqvrel 37911 df-funALTV 38008 df-disjALTV 38031 df-eldisj 38033 |
This theorem is referenced by: pet2 38176 |
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