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Theorem imadisjlnd 42912
Description: Natural deduction form of one negated side of imadisj 6080. (Contributed by Stanislas Polu, 9-Mar-2020.)
Hypothesis
Ref Expression
imadisjlnd.1 (𝜑 → (dom 𝐴𝐵) ≠ ∅)
Assertion
Ref Expression
imadisjlnd (𝜑 → (𝐴𝐵) ≠ ∅)

Proof of Theorem imadisjlnd
StepHypRef Expression
1 imadisjlnd.1 . 2 (𝜑 → (dom 𝐴𝐵) ≠ ∅)
2 imadisj 6080 . . . 4 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
32biimpi 215 . . 3 ((𝐴𝐵) = ∅ → (dom 𝐴𝐵) = ∅)
43necon3i 2974 . 2 ((dom 𝐴𝐵) ≠ ∅ → (𝐴𝐵) ≠ ∅)
51, 4syl 17 1 (𝜑 → (𝐴𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1542  wne 2941  cin 3948  c0 4323  dom cdm 5677  cima 5680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2155  ax-12 2172  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-nf 1787  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ne 2942  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-cnv 5685  df-dm 5687  df-rn 5688  df-res 5689  df-ima 5690
This theorem is referenced by:  wnefimgd  42913
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