Theorem wuncss

Description: The weak universe closure is a subset of any other weak universe containing the set. (Contributed by Mario Carneiro, 2-Jan-2017)

	Ref	Expression
Assertion	wuncss	⊢ ( ( 𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈 ) → ( wUniCl ‘ 𝐴 ) ⊆ 𝑈 )

Proof

Step	Hyp	Ref	Expression
1		ssexg	⊢ ( ( 𝐴 ⊆ 𝑈 ∧ 𝑈 ∈ WUni ) → 𝐴 ∈ V )
2	1	ancoms	⊢ ( ( 𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈 ) → 𝐴 ∈ V )
3		wuncval	⊢ ( 𝐴 ∈ V → ( wUniCl ‘ 𝐴 ) = ∩ { 𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢 } )
4	2 3	syl	⊢ ( ( 𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈 ) → ( wUniCl ‘ 𝐴 ) = ∩ { 𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢 } )
5		sseq2	⊢ ( 𝑢 = 𝑈 → ( 𝐴 ⊆ 𝑢 ↔ 𝐴 ⊆ 𝑈 ) )
6	5	intminss	⊢ ( ( 𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈 ) → ∩ { 𝑢 ∈ WUni ∣ 𝐴 ⊆ 𝑢 } ⊆ 𝑈 )
7	4 6	eqsstrd	⊢ ( ( 𝑈 ∈ WUni ∧ 𝐴 ⊆ 𝑈 ) → ( wUniCl ‘ 𝐴 ) ⊆ 𝑈 )