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Mirrors > Home > MPE Home > Th. List > Mathboxes > mainer | Structured version Visualization version GIF version |
Description: The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.) |
Ref | Expression |
---|---|
mainer | ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | eqvrelqseqdisj2 37291 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴) | |
2 | eldisjim 37246 | . . . 4 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴) |
4 | n0eldmqseq 37111 | . . . . 5 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) | |
5 | 4 | adantl 482 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴) |
6 | eldisjn0el 37268 | . . . . 5 ⊢ ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
7 | 1, 6 | syl 17 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (¬ ∅ ∈ 𝐴 ↔ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
8 | 5, 7 | mpbid 231 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (∪ 𝐴 / ∼ 𝐴) = 𝐴) |
9 | 3, 8 | jca 512 | . 2 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
10 | dferALTV2 37130 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
11 | dfcomember3 37136 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
12 | 9, 10, 11 | 3imtr4i 291 | 1 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 396 = wceq 1541 ∈ wcel 2106 ∅c0 4282 ∪ cuni 4865 dom cdm 5633 / cqs 8647 ∼ ccoels 36635 EqvRel weqvrel 36651 CoElEqvRel wcoeleqvrel 36653 ErALTV werALTV 36660 CoMembEr wcomember 36662 ElDisj weldisj 36670 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-10 2137 ax-11 2154 ax-12 2171 ax-ext 2707 ax-sep 5256 ax-nul 5263 ax-pr 5384 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-nf 1786 df-sb 2068 df-mo 2538 df-eu 2567 df-clab 2714 df-cleq 2728 df-clel 2814 df-nfc 2889 df-ne 2944 df-ral 3065 df-rex 3074 df-rmo 3353 df-rab 3408 df-v 3447 df-dif 3913 df-un 3915 df-in 3917 df-ss 3927 df-nul 4283 df-if 4487 df-sn 4587 df-pr 4589 df-op 4593 df-uni 4866 df-br 5106 df-opab 5168 df-id 5531 df-eprel 5537 df-xp 5639 df-rel 5640 df-cnv 5641 df-co 5642 df-dm 5643 df-rn 5644 df-res 5645 df-ima 5646 df-ec 8650 df-qs 8654 df-coss 36873 df-coels 36874 df-refrel 36974 df-cnvrefrel 36989 df-symrel 37006 df-trrel 37036 df-eqvrel 37047 df-coeleqvrel 37049 df-dmqs 37101 df-erALTV 37126 df-comember 37128 df-funALTV 37144 df-disjALTV 37167 df-eldisj 37169 |
This theorem is referenced by: partimcomember 37297 fences 37306 |
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