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Metamath Proof Explorer

Theorem intminss

Description: Under subset ordering, the intersection of a restricted class abstraction is less than or equal to any of its members. (Contributed by NM, 7-Sep-2013)

		Ref	Expression
	Hypothesis	intminss.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
	Assertion	intminss	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) → ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ⊆ 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		intminss.1	⊢ ( 𝑥 = 𝐴 → ( 𝜑 ↔ 𝜓 ) )
2	1	elrab	⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ↔ ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) )
3		intss1	⊢ ( 𝐴 ∈ { 𝑥 ∈ 𝐵 ∣ 𝜑 } → ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ⊆ 𝐴 )
4	2 3	sylbir	⊢ ( ( 𝐴 ∈ 𝐵 ∧ 𝜓 ) → ∩ { 𝑥 ∈ 𝐵 ∣ 𝜑 } ⊆ 𝐴 )