Theorem elpwuni 4053

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

Assertion

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

Proof of Theorem elpwuni

Step	Hyp	Ref	Expression
1		sspwuni 4051	. 2 ⊢ (A ⊆ ℘B ↔ ∪A ⊆ B)
2		unissel 3920	. . . 4 ⊢ ((∪A ⊆ B ∧ B ∈ A) → ∪A = B)
3	2	expcom 424	. . 3 ⊢ (B ∈ A → (∪A ⊆ B → ∪A = B))
4		eqimss 3323	. . 3 ⊢ (∪A = B → ∪A ⊆ B)
5	3, 4	impbid1 194	. 2 ⊢ (B ∈ A → (∪A ⊆ B ↔ ∪A = B))
6	1, 5	syl5bb 248	1 ⊢ (B ∈ A → (A ⊆ ℘B ↔ ∪A = B))

Syntax hints:   → wi 4   ↔ wb 176   = wceq 1642   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722  ∪cuni 3891

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-ral 2619  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724  df-uni 3892

This theorem is referenced by: (None)