Theorem ssdisjd

Description: Subset preserves disjointness. Deduction form of ssdisj . (Contributed by Zhi Wang, 7-Sep-2024)

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

Proof

Step	Hyp	Ref	Expression
1		ssdisjd.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		ssdisjd.2	⊢ ( 𝜑 → ( 𝐵 ∩ 𝐶 ) = ∅ )
3	1	ssrind	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) ⊆ ( 𝐵 ∩ 𝐶 ) )
4		sseq0	⊢ ( ( ( 𝐴 ∩ 𝐶 ) ⊆ ( 𝐵 ∩ 𝐶 ) ∧ ( 𝐵 ∩ 𝐶 ) = ∅ ) → ( 𝐴 ∩ 𝐶 ) = ∅ )
5	3 2 4	syl2anc	⊢ ( 𝜑 → ( 𝐴 ∩ 𝐶 ) = ∅ )