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Theorem imadisjld 42374
Description: Natural dduction form of one side of imadisj 6030. (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 6030 . 2 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
31, 2sylibr 233 1 (𝜑 → (𝐴𝐵) = ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1541  cin 3907  c0 4280  dom cdm 5631  cima 5634
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-10 2137  ax-11 2154  ax-12 2171  ax-ext 2707  ax-sep 5254  ax-nul 5261  ax-pr 5382
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-nf 1786  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3063  df-rex 3072  df-rab 3406  df-v 3445  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4281  df-if 4485  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-opab 5166  df-xp 5637  df-cnv 5639  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644
This theorem is referenced by: (None)
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