Users' Mathboxes Mathbox for Peter Mazsa < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  mainer Structured version   Visualization version   GIF version

Theorem mainer 38215
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 38210 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴)
2 eldisjim 38165 . . . 4 ( ElDisj 𝐴 → CoElEqvRel 𝐴)
31, 2syl 17 . . 3 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴)
4 n0eldmqseq 38030 . . . . 5 ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
54adantl 481 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴)
6 eldisjn0el 38187 . . . . 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 38049 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
11 dfcomember3 38055 . 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 1533  wcel 2098  c0 4317   cuni 4902  dom cdm 5669   / cqs 8701  ccoels 37555   EqvRel weqvrel 37571   CoElEqvRel wcoeleqvrel 37573   ErALTV werALTV 37580   CoMembEr wcomember 37582   ElDisj weldisj 37590
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-10 2129  ax-11 2146  ax-12 2163  ax-ext 2697  ax-sep 5292  ax-nul 5299  ax-pr 5420
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-nf 1778  df-sb 2060  df-mo 2528  df-eu 2557  df-clab 2704  df-cleq 2718  df-clel 2804  df-nfc 2879  df-ne 2935  df-ral 3056  df-rex 3065  df-rmo 3370  df-rab 3427  df-v 3470  df-dif 3946  df-un 3948  df-in 3950  df-ss 3960  df-nul 4318  df-if 4524  df-sn 4624  df-pr 4626  df-op 4630  df-uni 4903  df-br 5142  df-opab 5204  df-id 5567  df-eprel 5573  df-xp 5675  df-rel 5676  df-cnv 5677  df-co 5678  df-dm 5679  df-rn 5680  df-res 5681  df-ima 5682  df-ec 8704  df-qs 8708  df-coss 37792  df-coels 37793  df-refrel 37893  df-cnvrefrel 37908  df-symrel 37925  df-trrel 37955  df-eqvrel 37966  df-coeleqvrel 37968  df-dmqs 38020  df-erALTV 38045  df-comember 38047  df-funALTV 38063  df-disjALTV 38086  df-eldisj 38088
This theorem is referenced by:  partimcomember  38216  fences  38225
  Copyright terms: Public domain W3C validator