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Metamath Proof Explorer

Theorem ssind

Description: A deduction showing that a subclass of two classes is a subclass of their intersection. (Contributed by Jonathan Ben-Naim, 3-Jun-2011)

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

Proof

Step	Hyp	Ref	Expression
1		ssind.1	⊢ ( 𝜑 → 𝐴 ⊆ 𝐵 )
2		ssind.2	⊢ ( 𝜑 → 𝐴 ⊆ 𝐶 )
3	1 2	jca	⊢ ( 𝜑 → ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 ) )
4		ssin	⊢ ( ( 𝐴 ⊆ 𝐵 ∧ 𝐴 ⊆ 𝐶 ) ↔ 𝐴 ⊆ ( 𝐵 ∩ 𝐶 ) )
5	3 4	sylib	⊢ ( 𝜑 → 𝐴 ⊆ ( 𝐵 ∩ 𝐶 ) )