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Theorem imadisjld 44173
Description: Natural dduction form of one side of imadisj 6098. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypothesis
Ref Expression
imadisjld.1 (𝜑 → (dom 𝐴𝐵) = ∅)
Assertion
Ref Expression
imadisjld (𝜑 → (𝐴𝐵) = ∅)

Proof of Theorem imadisjld
StepHypRef Expression
1 imadisjld.1 . 2 (𝜑 → (dom 𝐴𝐵) = ∅)
2 imadisj 6098 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
31, 2sylibr 234 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1540  cin 3950  c0 4333  dom cdm 5685  cima 5688
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-xp 5691  df-cnv 5693  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698
This theorem is referenced by: (None)
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