Theorem elpwuni 5076

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

Assertion

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

Proof of Theorem elpwuni

Step	Hyp	Ref	Expression
1		sspwuni 5071	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2		unissel 4910	. . . 4 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵)
3	2	expcom 419	. . 3 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 → ∪ 𝐴 = 𝐵))
4		eqimss 3998	. . 3 ⊢ (∪ 𝐴 = 𝐵 → ∪ 𝐴 ⊆ 𝐵)
5	3, 4	impbid1 228	. 2 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 ↔ ∪ 𝐴 = 𝐵))
6	1, 5	bitrid 286	1 ⊢ (𝐵 ∈ 𝐴 → (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 = 𝐵))

This proof depends on syntax axioms:   → wi 4   ↔ wb 209   = wceq 1570   ∈ wcel 2146   ⊆ wss 3908  𝒫 cpw 4567  ∪ cuni 4877

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-v 3460  df-ss 3925  df-pw 4569  df-uni 4878

This theorem is used by:  mreuni  17677  ustuni  24420  utopbas  24429  issgon  34544  br2base  34691