Theorem imadisjld 45160

Description: Natural dduction form of one side of imadisj 6076. (Contributed by Stanislas Polu, 9-Mar-2020.)

Hypothesis

Ref	Expression
imadisjld.1	⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
imadisjld	⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)

Proof of Theorem imadisjld

Step	Hyp	Ref	Expression
1		imadisjld.1	. 2 ⊢ (𝜑 → (dom 𝐴 ∩ 𝐵) = ∅)
2		imadisj 6076	. 2 ⊢ ((𝐴 “ 𝐵) = ∅ ↔ (dom 𝐴 ∩ 𝐵) = ∅)
3	1, 2	sylibr 237	1 ⊢ (𝜑 → (𝐴 “ 𝐵) = ∅)

This proof depends on syntax axioms:   → wi 4   = wceq 1569   ∩ cin 4089  ∅c0 4278  dom cdm 5650   “ cima 5653

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146  ax-9 2154  ax-ext 2732  ax-sep 5248  ax-pr 5390

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3an 1105  df-tru 1572  df-fal 1582  df-ex 1812  df-sb 2099  df-clab 2739  df-cleq 2752  df-clel 2835  df-ral 3077  df-rex 3087  df-rab 3413  df-v 3452  df-ss 3896  df-dif 4047  df-in 4091  df-un 4151  df-nul 4279  df-if 4482  df-sn 4584  df-pr 4586  df-op 4590  df-br 5103  df-opab 5167  df-xp 5656  df-cnv 5658  df-dm 5660  df-rn 5661  df-res 5662  df-ima 5663

This theorem is used by: (None)