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Theorem dfeldisj2 38719
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 38708 . 2 ( ElDisj 𝐴 ↔ Disj ( E ↾ 𝐴))
2 relres 6023 . . 3 Rel ( E ↾ 𝐴)
3 dfdisjALTV2 38715 . . 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 3951   I cid 5577   E cep 5583  ccnv 5684  cres 5687  Rel wrel 5690  ccoss 38182   Disj wdisjALTV 38216   ElDisj weldisj 38218
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-11 2157  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-coss 38412  df-cnvrefrel 38528  df-disjALTV 38706  df-eldisj 38708
This theorem is referenced by: (None)
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