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Theorem dfeldisj2 38700
Description: Alternate definition of the disjoint elementhood predicate. (Contributed by Peter Mazsa, 19-Sep-2021.)
Assertion
Ref Expression
dfeldisj2 ( ElDisj 𝐴 ↔ ≀ ( E ↾ 𝐴) ⊆ I )

Proof of Theorem dfeldisj2
StepHypRef Expression
1 df-eldisj 38689 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
2 relres 6026 . . 3 Rel ( E ↾ 𝐴)
3 dfdisjALTV2 38696 . . 3 ( Disj ( E ↾ 𝐴) ↔ ( ≀ ( E ↾ 𝐴) ⊆ I ∧ Rel ( E ↾ 𝐴)))
42, 3mpbiran2 710 . 2 ( Disj ( E ↾ 𝐴) ↔ ≀ ( E ↾ 𝐴) ⊆ I )
51, 4bitri 275 1 ( ElDisj 𝐴 ↔ ≀ ( E ↾ 𝐴) ⊆ I )
Colors of variables: wff setvar class
Syntax hints:  wb 206  wss 3963   I cid 5582   E cep 5588  ccnv 5688  cres 5691  Rel wrel 5694  ccoss 38162   Disj wdisjALTV 38196   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-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  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-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-res 5701  df-coss 38393  df-cnvrefrel 38509  df-disjALTV 38687  df-eldisj 38689
This theorem is referenced by: (None)
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