Theorem ssintub

Description: Subclass of the least upper bound. (Contributed by NM, 8-Aug-2000)

	Ref	Expression
Assertion	ssintub	⊢ 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 }

Proof

Step	Hyp	Ref	Expression
1		ssint	⊢ ( 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } ↔ ∀ 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } 𝐴 ⊆ 𝑦 )
2		sseq2	⊢ ( 𝑥 = 𝑦 → ( 𝐴 ⊆ 𝑥 ↔ 𝐴 ⊆ 𝑦 ) )
3	2	elrab	⊢ ( 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } ↔ ( 𝑦 ∈ 𝐵 ∧ 𝐴 ⊆ 𝑦 ) )
4	3	simprbi	⊢ ( 𝑦 ∈ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 } → 𝐴 ⊆ 𝑦 )
5	1 4	mprgbir	⊢ 𝐴 ⊆ ∩ { 𝑥 ∈ 𝐵 ∣ 𝐴 ⊆ 𝑥 }