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Theorem imadisjlnd 38285
Description: Natural deduction form of one negated side of imadisj 5482. (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 5482 . . . 4 ((𝐴𝐵) = ∅ ↔ (dom 𝐴𝐵) = ∅)
32biimpi 206 . . 3 ((𝐴𝐵) = ∅ → (dom 𝐴𝐵) = ∅)
43necon3i 2825 . 2 ((dom 𝐴𝐵) ≠ ∅ → (𝐴𝐵) ≠ ∅)
51, 4syl 17 1 (𝜑 → (𝐴𝐵) ≠ ∅)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1482  wne 2793  cin 3571  c0 3913  dom cdm 5112  cima 5115
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1721  ax-4 1736  ax-5 1838  ax-6 1887  ax-7 1934  ax-9 1998  ax-10 2018  ax-11 2033  ax-12 2046  ax-13 2245  ax-ext 2601  ax-sep 4779  ax-nul 4787  ax-pr 4904
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1485  df-ex 1704  df-nf 1709  df-sb 1880  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2752  df-ne 2794  df-ral 2916  df-rex 2917  df-rab 2920  df-v 3200  df-dif 3575  df-un 3577  df-in 3579  df-ss 3586  df-nul 3914  df-if 4085  df-sn 4176  df-pr 4178  df-op 4182  df-br 4652  df-opab 4711  df-xp 5118  df-cnv 5120  df-dm 5122  df-rn 5123  df-res 5124  df-ima 5125
This theorem is referenced by: (None)
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