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| Mirrors > Home > MPE Home > Th. List > Mathboxes > eqvrelqseqdisj3 | Structured version Visualization version GIF version | ||
| Description: Implication of eqvreldisj3 38828, lemma for the Member Partition Equivalence Theorem mpet3 38838. (Contributed by Peter Mazsa, 27-Oct-2020.) (Revised by Peter Mazsa, 24-Sep-2021.) | 
| Ref | Expression | 
|---|---|
| eqvrelqseqdisj3 | ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ 𝐴)) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | eqvreldisj3 38828 | . . 3 ⊢ ( EqvRel 𝑅 → Disj (◡ E ↾ (𝐵 / 𝑅))) | |
| 2 | 1 | adantr 480 | . 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ (𝐵 / 𝑅))) | 
| 3 | reseq2 5991 | . . . 4 ⊢ ((𝐵 / 𝑅) = 𝐴 → (◡ E ↾ (𝐵 / 𝑅)) = (◡ E ↾ 𝐴)) | |
| 4 | 3 | disjeqd 38738 | . . 3 ⊢ ((𝐵 / 𝑅) = 𝐴 → ( Disj (◡ E ↾ (𝐵 / 𝑅)) ↔ Disj (◡ E ↾ 𝐴))) | 
| 5 | 4 | adantl 481 | . 2 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → ( Disj (◡ E ↾ (𝐵 / 𝑅)) ↔ Disj (◡ E ↾ 𝐴))) | 
| 6 | 2, 5 | mpbid 232 | 1 ⊢ (( EqvRel 𝑅 ∧ (𝐵 / 𝑅) = 𝐴) → Disj (◡ E ↾ 𝐴)) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1539 E cep 5582 ◡ccnv 5683 ↾ cres 5686 / cqs 8745 EqvRel weqvrel 38200 Disj wdisjALTV 38217 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-10 2140 ax-11 2156 ax-12 2176 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-nf 1783 df-sb 2064 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2728 df-clel 2815 df-nfc 2891 df-ne 2940 df-ral 3061 df-rex 3070 df-rmo 3379 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-in 3957 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-id 5577 df-eprel 5583 df-xp 5690 df-rel 5691 df-cnv 5692 df-co 5693 df-dm 5694 df-rn 5695 df-res 5696 df-ima 5697 df-ec 8748 df-qs 8752 df-coss 38413 df-refrel 38514 df-cnvrefrel 38529 df-symrel 38546 df-trrel 38576 df-eqvrel 38587 df-funALTV 38684 df-disjALTV 38707 df-eldisj 38709 | 
| This theorem is referenced by: eqvrelqseqdisj4 38834 eqvrelqseqdisj5 38835 mpet3 38838 cpet2 38839 | 
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