Theorem imadisjlnd 37479

Description: Natural deduction form of one negated side of imadisj 5403. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypothesis

Ref	Expression
imadisjlnd.1	⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅)

Assertion

Ref	Expression
imadisjlnd	⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅)

Proof of Theorem imadisjlnd

Step	Hyp	Ref	Expression
1		imadisjlnd.1	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) ≠ ∅)
2		imadisj 5403	. . . 4 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
3	2	biimpi 205	. . 3 ⊢ ((𝐴 “ 𝐵) = ∅ → (dom 𝐴 ∩ 𝐵) = ∅)
4	3	necon3i 2814	. 2 ⊢ ((dom 𝐴 ∩ 𝐵) ≠ ∅ → (𝐴 “ 𝐵) ≠ ∅)
5	1, 4	syl 17	1 ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅)

Syntax hints:   → wi 4   = wceq 1475   ≠ wne 2780   ∩ cin 3539  ∅c0 3874  dom cdm 5038   “ cima 5041

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ne 2782  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-opab 4644  df-xp 5044  df-cnv 5046  df-dm 5048  df-rn 5049  df-res 5050  df-ima 5051

This theorem is referenced by: (None)