Theorem ssunieq

Description: Relationship implying union. (Contributed by NM, 10-Nov-1999)

	Ref	Expression
Assertion	ssunieq	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 ) → 𝐴 = ∪ 𝐵 )

Proof

Step	Hyp	Ref	Expression
1		elssuni	⊢ ( 𝐴 ∈ 𝐵 → 𝐴 ⊆ ∪ 𝐵 )
2		unissb	⊢ ( ∪ 𝐵 ⊆ 𝐴 ↔ ∀ 𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 )
3	2	biimpri	⊢ ( ∀ 𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 → ∪ 𝐵 ⊆ 𝐴 )
4	1 3	anim12i	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 ) → ( 𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴 ) )
5		eqss	⊢ ( 𝐴 = ∪ 𝐵 ↔ ( 𝐴 ⊆ ∪ 𝐵 ∧ ∪ 𝐵 ⊆ 𝐴 ) )
6	4 5	sylibr	⊢ ( ( 𝐴 ∈ 𝐵 ∧ ∀ 𝑥 ∈ 𝐵 𝑥 ⊆ 𝐴 ) → 𝐴 = ∪ 𝐵 )