Theorem ssdisjdr 46042

Description: Subset preserves disjointness. Deduction form of ssdisj 4390. Alternatively this could be proved with ineqcom 4133 in tandem with ssdisjd 46041. (Contributed by Zhi Wang, 7-Sep-2024.)

Hypotheses

Ref	Expression
ssdisjd.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
ssdisjdr.2	⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅)

Assertion

Ref	Expression
ssdisjdr	⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅)

Proof of Theorem ssdisjdr

Step	Hyp	Ref	Expression
1		ssdisjd.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2		sslin 4165	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵))
3	1, 2	syl 17	. 2 ⊢ (𝜑 → (𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵))
4		ssdisjdr.2	. 2 ⊢ (𝜑 → (𝐶 ∩ 𝐵) = ∅)
5		sseq0 4330	. 2 ⊢ (((𝐶 ∩ 𝐴) ⊆ (𝐶 ∩ 𝐵) ∧ (𝐶 ∩ 𝐵) = ∅) → (𝐶 ∩ 𝐴) = ∅)
6	3, 4, 5	syl2anc 583	1 ⊢ (𝜑 → (𝐶 ∩ 𝐴) = ∅)

Syntax hints:   → wi 4   = wceq 1539   ∩ cin 3882   ⊆ wss 3883  ∅c0 4253

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-rab 3072  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254

This theorem is referenced by:  predisj  46044  seposep  46107