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Theorem mainer 37296
Description: The Main Theorem of Equivalences: every equivalence relation implies equivalent comembers. (Contributed by Peter Mazsa, 26-Sep-2021.)
Assertion
Ref Expression
mainer (𝑅 ErALTV 𝐴 → CoMembEr 𝐴)

Proof of Theorem mainer
StepHypRef Expression
1 eqvrelqseqdisj2 37291 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴)
2 eldisjim 37246 . . . 4 ( ElDisj 𝐴 → CoElEqvRel 𝐴)
31, 2syl 17 . . 3 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴)
4 n0eldmqseq 37111 . . . . 5 ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
54adantl 482 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴)
6 eldisjn0el 37268 . . . . 5 ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴))
71, 6syl 17 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴))
85, 7mpbid 231 . . 3 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ( 𝐴 /𝐴) = 𝐴)
93, 8jca 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))
129, 10, 113imtr4i 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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