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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 38301 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴) | |
2 | eldisjim 38256 | . . . 4 ⊢ ( ElDisj 𝐴 → CoElEqvRel 𝐴) | |
3 | 1, 2 | syl 17 | . . 3 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴) |
4 | n0eldmqseq 38121 | . . . . 5 ⊢ ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴) | |
5 | 4 | adantl 481 | . . . 4 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴) |
6 | eldisjn0el 38278 | . . . . 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 511 | . 2 ⊢ (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( CoElEqvRel 𝐴 ∧ (∪ 𝐴 / ∼ 𝐴) = 𝐴)) |
10 | dferALTV2 38140 | . 2 ⊢ (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴)) | |
11 | dfcomember3 38146 | . 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 205 ∧ wa 395 = wceq 1534 ∈ wcel 2099 ∅c0 4323 ∪ cuni 4908 dom cdm 5678 / cqs 8724 ∼ ccoels 37649 EqvRel weqvrel 37665 CoElEqvRel wcoeleqvrel 37667 ErALTV werALTV 37674 CoMembEr wcomember 37676 ElDisj weldisj 37684 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-sep 5299 ax-nul 5306 ax-pr 5429 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-nfc 2881 df-ne 2938 df-ral 3059 df-rex 3068 df-rmo 3373 df-rab 3430 df-v 3473 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4324 df-if 4530 df-sn 4630 df-pr 4632 df-op 4636 df-uni 4909 df-br 5149 df-opab 5211 df-id 5576 df-eprel 5582 df-xp 5684 df-rel 5685 df-cnv 5686 df-co 5687 df-dm 5688 df-rn 5689 df-res 5690 df-ima 5691 df-ec 8727 df-qs 8731 df-coss 37883 df-coels 37884 df-refrel 37984 df-cnvrefrel 37999 df-symrel 38016 df-trrel 38046 df-eqvrel 38057 df-coeleqvrel 38059 df-dmqs 38111 df-erALTV 38136 df-comember 38138 df-funALTV 38154 df-disjALTV 38177 df-eldisj 38179 |
This theorem is referenced by: partimcomember 38307 fences 38316 |
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