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Theorem imadisjld 44150
Description: Natural dduction form of one side of imadisj 6100. (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 6100 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
31, 2sylibr 234 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  cin 3962  c0 4339  dom cdm 5689  cima 5692
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-xp 5695  df-cnv 5697  df-dm 5699  df-rn 5700  df-res 5701  df-ima 5702
This theorem is referenced by: (None)
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