Theorem elpwuni 5065

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 5060	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2		unissel 4900	. . . 4 ⊢ ((∪ 𝐴 ⊆ 𝐵 ∧ 𝐵 ∈ 𝐴) → ∪ 𝐴 = 𝐵)
3	2	expcom 419	. . 3 ⊢ (𝐵 ∈ 𝐴 → (∪ 𝐴 ⊆ 𝐵 → ∪ 𝐴 = 𝐵))
4		eqimss 3989	. . 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 2145   ⊆ wss 3899  𝒫 cpw 4557  ∪ cuni 4867

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 2147  ax-9 2155  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-v 3453  df-ss 3916  df-pw 4559  df-uni 4868

This theorem is used by:  mreuni  17750  ustuni  24525  utopbas  24534  issgon  34737  br2base  34884