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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 38811 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴) | |
2 | eldisjim 38766 | . . . 4 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴) |
4 | n0eldmqseq 38631 | . . . . 5 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) | |
5 | 4 | adantl 481 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴) |
6 | eldisjn0el 38788 | . . . . 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 232 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (∪ 𝐴 / ∼ 𝐴) = 𝐴) |
9 | 3, 8 | jca 511 | . 2 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
10 | dferALTV2 38650 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
11 | dfcomember3 38656 | . 2 ⊢ ( CoMembEr 𝐴 ↔ ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) | |
12 | 9, 10, 11 | 3imtr4i 292 | 1 ⊢ (𝑅 ErALTV 𝐴 → CoMembEr 𝐴) |
Colors of variables: wff setvar class |
Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 = wceq 1537 ∈ wcel 2106 ∅c0 4339 ∪ cuni 4912 dom cdm 5689 / cqs 8743 ∼ ccoels 38163 EqvRel weqvrel 38179 CoElEqvRel wcoeleqvrel 38181 ErALTV werALTV 38188 CoMembEr wcomember 38190 ElDisj weldisj 38198 |
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 |
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-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-ec 8746 df-qs 8750 df-coss 38393 df-coels 38394 df-refrel 38494 df-cnvrefrel 38509 df-symrel 38526 df-trrel 38556 df-eqvrel 38567 df-coeleqvrel 38569 df-dmqs 38621 df-erALTV 38646 df-comember 38648 df-funALTV 38664 df-disjALTV 38687 df-eldisj 38689 |
This theorem is referenced by: partimcomember 38817 fences 38826 |
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