Theorem imadisjlnd 6042

Description: Deduction form of one negated side of imadisj 6041. (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 6041	. . . 4 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
3	2	biimpi 216	. . 3 ⊢ ((𝐴 “ 𝐵) = ∅ → (dom 𝐴 ∩ 𝐵) = ∅)
4	3	necon3i 2965	. 2 ⊢ ((dom 𝐴 ∩ 𝐵) ≠ ∅ → (𝐴 “ 𝐵) ≠ ∅)
5	1, 4	syl 17	1 ⊢ (𝜑 → (𝐴 “ 𝐵) ≠ ∅)

Syntax hints:   → wi 4   = wceq 1542   ≠ wne 2933   ∩ cin 3889  ∅c0 4274  dom cdm 5626   “ cima 5629

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709  ax-sep 5232  ax-pr 5372

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-ne 2934  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5632  df-cnv 5634  df-dm 5636  df-rn 5637  df-res 5638  df-ima 5639

This theorem is referenced by:  vonf1owev  35310  weiunfrlem  36666  wnefimgd  44610  grimuhgr  48379