Theorem elpwuni

Description: Relationship for power class and union. (Contributed by NM, 18-Jul-2006)

	Ref	Expression
Assertion	elpwuni	⊢ ( 𝐵 ∈ 𝐴 → ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵 ) )

Proof

Step	Hyp	Ref	Expression
1		sspwuni	⊢ ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵 )
2		unissel	⊢ ( ( ∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴 ) → ∪ 𝐴 = 𝐵 )
3	2	expcom	⊢ ( 𝐵 ∈ 𝐴 → ( ∪ 𝐴 ⊆ 𝐵 → ∪ 𝐴 = 𝐵 ) )
4		eqimss	⊢ ( ∪ 𝐴 = 𝐵 → ∪ 𝐴 ⊆ 𝐵 )
5	3 4	impbid1	⊢ ( 𝐵 ∈ 𝐴 → ( ∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 = 𝐵 ) )
6	1 5	bitrid	⊢ ( 𝐵 ∈ 𝐴 → ( 𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵 ) )