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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 37687 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴) | |
2 | eldisjim 37642 | . . . 4 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴) |
4 | n0eldmqseq 37507 | . . . . 5 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) | |
5 | 4 | adantl 482 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴) |
6 | eldisjn0el 37664 | . . . . 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 37526 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
11 | dfcomember3 37532 | . 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 4321 ∪ cuni 4907 dom cdm 5675 / cqs 8698 ∼ ccoels 37032 EqvRel weqvrel 37048 CoElEqvRel wcoeleqvrel 37050 ErALTV werALTV 37057 CoMembEr wcomember 37059 ElDisj weldisj 37067 |
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 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426 |
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 2534 df-eu 2563 df-clab 2710 df-cleq 2724 df-clel 2810 df-nfc 2885 df-ne 2941 df-ral 3062 df-rex 3071 df-rmo 3376 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-uni 4908 df-br 5148 df-opab 5210 df-id 5573 df-eprel 5579 df-xp 5681 df-rel 5682 df-cnv 5683 df-co 5684 df-dm 5685 df-rn 5686 df-res 5687 df-ima 5688 df-ec 8701 df-qs 8705 df-coss 37269 df-coels 37270 df-refrel 37370 df-cnvrefrel 37385 df-symrel 37402 df-trrel 37432 df-eqvrel 37443 df-coeleqvrel 37445 df-dmqs 37497 df-erALTV 37522 df-comember 37524 df-funALTV 37540 df-disjALTV 37563 df-eldisj 37565 |
This theorem is referenced by: partimcomember 37693 fences 37702 |
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