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 38816
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 38811 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ElDisj 𝐴)
2 eldisjim 38766 . . . 4 ( ElDisj 𝐴 → CoElEqvRel 𝐴)
31, 2syl 17 . . 3 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → CoElEqvRel 𝐴)
4 n0eldmqseq 38631 . . . . 5 ((dom 𝑅 / 𝑅) = 𝐴 → ¬ ∅ ∈ 𝐴)
54adantl 481 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → ¬ ∅ ∈ 𝐴)
6 eldisjn0el 38788 . . . . 5 ( ElDisj 𝐴 → (¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴))
71, 6syl 17 . . . 4 (( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴) → (¬ ∅ ∈ 𝐴 ↔ ( 𝐴 /𝐴) = 𝐴))
85, 7mpbid 232 . . 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 38650 . 2 (𝑅 ErALTV 𝐴 ↔ ( EqvRel 𝑅 ∧ (dom 𝑅 / 𝑅) = 𝐴))
11 dfcomember3 38656 . 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 206  wa 395   = wceq 1537  wcel 2106  c0 4339   cuni 4912  dom cdm 5689   / cqs 8743  ccoels 38163   EqvRel weqvrel 38179   CoElEqvRel wcoeleqvrel 38181   ErALTV werALTV 38188   CoMembEr wcomember 38190   ElDisj weldisj 38198
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-10 2139  ax-11 2155  ax-12 2175  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-nf 1781  df-sb 2063  df-mo 2538  df-eu 2567  df-clab 2713  df-cleq 2727  df-clel 2814  df-nfc 2890  df-ne 2939  df-ral 3060  df-rex 3069  df-rmo 3378  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-in 3970  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-uni 4913  df-br 5149  df-opab 5211  df-id 5583  df-eprel 5589  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702  df-ec 8746  df-qs 8750  df-coss 38393  df-coels 38394  df-refrel 38494  df-cnvrefrel 38509  df-symrel 38526  df-trrel 38556  df-eqvrel 38567  df-coeleqvrel 38569  df-dmqs 38621  df-erALTV 38646  df-comember 38648  df-funALTV 38664  df-disjALTV 38687  df-eldisj 38689
This theorem is referenced by:  partimcomember  38817  fences  38826
  Copyright terms: Public domain W3C validator