Theorem clss2lem

Description: The closure of a property is a superset of the closure of a less restrictive property. (Contributed by RP, 24-Jul-2020)

	Ref	Expression
Hypothesis	clss2lem.1	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
Assertion	clss2lem	⊢ ( 𝜑 → ∩ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) } ⊆ ∩ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) } )

Proof

Step	Hyp	Ref	Expression
1		clss2lem.1	⊢ ( 𝜑 → ( 𝜒 → 𝜓 ) )
2	1	adantld	⊢ ( 𝜑 → ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → 𝜓 ) )
3	2	alrimiv	⊢ ( 𝜑 → ∀ 𝑥 ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → 𝜓 ) )
4		pm5.3	⊢ ( ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → 𝜓 ) ↔ ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) ) )
5	4	albii	⊢ ( ∀ 𝑥 ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → 𝜓 ) ↔ ∀ 𝑥 ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) ) )
6		ss2ab	⊢ ( { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) } ⊆ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) } ↔ ∀ 𝑥 ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) ) )
7	5 6	bitr4i	⊢ ( ∀ 𝑥 ( ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) → 𝜓 ) ↔ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) } ⊆ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) } )
8	3 7	sylib	⊢ ( 𝜑 → { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) } ⊆ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) } )
9		intss	⊢ ( { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) } ⊆ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) } → ∩ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) } ⊆ ∩ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) } )
10	8 9	syl	⊢ ( 𝜑 → ∩ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜓 ) } ⊆ ∩ { 𝑥 ∣ ( 𝑋 ⊆ 𝑥 ∧ 𝜒 ) } )