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Theorem imadisjlnd 39230
Description: Natural deduction form of one negated side of imadisj 5700. (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 5700 . . . 4 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
32biimpi 208 . . 3 ((𝐴𝐵) = ∅ → (dom 𝐴𝐵) = ∅)
43necon3i 3002 . 2 ((dom 𝐴𝐵) ≠ ∅ → (𝐴𝐵) ≠ ∅)
51, 4syl 17 1 (𝜑 → (𝐴𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1653  wne 2970  cin 3767  c0 4114  dom cdm 5311  cima 5314
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2776  ax-sep 4974  ax-nul 4982  ax-pr 5096
This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2785  df-cleq 2791  df-clel 2794  df-nfc 2929  df-ne 2971  df-ral 3093  df-rex 3094  df-rab 3097  df-v 3386  df-dif 3771  df-un 3773  df-in 3775  df-ss 3782  df-nul 4115  df-if 4277  df-sn 4368  df-pr 4370  df-op 4374  df-br 4843  df-opab 4905  df-xp 5317  df-cnv 5319  df-dm 5321  df-rn 5322  df-res 5323  df-ima 5324
This theorem is referenced by: (None)
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