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Theorem mainer 37692
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 37687 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴)
2 eldisjim 37642 . . . 4 ( ElDisj 𝐴 → CoElEqvRel 𝐴)
31, 2syl 17 . . 3 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴)
4 n0eldmqseq 37507 . . . . 5 ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
54adantl 482 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴)
6 eldisjn0el 37664 . . . . 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 37526 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
11 dfcomember3 37532 . 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 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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