Theorem disjdifg 4431

Description: A class does not intersect a relative complement of a superclass. (Contributed by NM, 24-Mar-1998.) Generalize from disjdif 4432. (Revised by BJ, 19-Jul-2026.)

Assertion

Ref	Expression
disjdifg	⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 ∖ 𝐵)) = ∅)

Proof of Theorem disjdifg

Step	Hyp	Ref	Expression
1		ssinss1 4197	. 2 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ 𝐶) ⊆ 𝐵)
2		inssdif0 4328	. 2 ⊢ ((𝐴 ∩ 𝐶) ⊆ 𝐵 ↔ (𝐴 ∩ (𝐶 ∖ 𝐵)) = ∅)
3	1, 2	sylib 221	1 ⊢ (𝐴 ⊆ 𝐵 → (𝐴 ∩ (𝐶 ∖ 𝐵)) = ∅)

Syntax hints:   → wi 4   = wceq 1568   ∖ cdif 3901   ∩ cin 3903   ⊆ wss 3904  ∅c0 4285

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1823  ax-4 1837  ax-5 1938  ax-6 1995  ax-7 2036  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 210  df-an 401  df-tru 1571  df-fal 1581  df-ex 1808  df-sb 2095  df-clab 2740  df-cleq 2753  df-clel 2836  df-rab 3415  df-v 3455  df-dif 3907  df-in 3911  df-ss 3921  df-nul 4286

This theorem is referenced by:  disjdif  4432