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Theorem imadisjlnd 6082
Description: Deduction form of one negated side of imadisj 6081. (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 6081 . . . 4 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
32biimpi 215 . . 3 ((𝐴𝐵) = ∅ → (dom 𝐴𝐵) = ∅)
43necon3i 2963 . 2 ((dom 𝐴𝐵) ≠ ∅ → (𝐴𝐵) ≠ ∅)
51, 4syl 17 1 (𝜑 → (𝐴𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1534  wne 2930  cin 3947  c0 4324  dom cdm 5674  cima 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-10 2130  ax-11 2147  ax-12 2167  ax-ext 2697  ax-sep 5296  ax-nul 5303  ax-pr 5425
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 846  df-3an 1086  df-tru 1537  df-fal 1547  df-ex 1775  df-nf 1779  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ne 2931  df-ral 3052  df-rex 3061  df-rab 3421  df-v 3466  df-dif 3951  df-un 3953  df-in 3955  df-ss 3965  df-nul 4325  df-if 4526  df-sn 4626  df-pr 4628  df-op 4632  df-br 5146  df-opab 5208  df-xp 5680  df-cnv 5682  df-dm 5684  df-rn 5685  df-res 5686  df-ima 5687
This theorem is referenced by:  wnefimgd  43864  grimuhgr  47492
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