Theorem sspwuni 4051

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

Assertion

Ref	Expression
sspwuni	⊢ (A ⊆ ℘B ↔ ∪A ⊆ B)

Proof of Theorem sspwuni

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		vex 2862	. . . 4 ⊢ x ∈ V
2	1	elpw 3728	. . 3 ⊢ (x ∈ ℘B ↔ x ⊆ B)
3	2	ralbii 2638	. 2 ⊢ (∀x ∈ A x ∈ ℘B ↔ ∀x ∈ A x ⊆ B)
4		dfss3 3263	. 2 ⊢ (A ⊆ ℘B ↔ ∀x ∈ A x ∈ ℘B)
5		unissb 3921	. 2 ⊢ (∪A ⊆ B ↔ ∀x ∈ A x ⊆ B)
6	3, 4, 5	3bitr4i 268	1 ⊢ (A ⊆ ℘B ↔ ∪A ⊆ B)

Syntax hints:   ↔ wb 176   ∈ wcel 1710  ∀wral 2614   ⊆ 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:  pwssb  4052  elpwuni  4053  rintn0  4056  qsss  5986