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Theorem mainer 38306
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 38301 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴)
2 eldisjim 38256 . . . 4 ( ElDisj 𝐴 → CoElEqvRel 𝐴)
31, 2syl 17 . . 3 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴)
4 n0eldmqseq 38121 . . . . 5 ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
54adantl 481 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴)
6 eldisjn0el 38278 . . . . 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 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))
129, 10, 113imtr4i 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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