Theorem sspwuni 4052

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 2863	. . . 4 ⊢ x ∈ V
2	1	elpw 3729	. . 3 ⊢ (x ∈ ℘B ↔ x ⊆ B)
3	2	ralbii 2639	. 2 ⊢ (∀x ∈ A x ∈ ℘B ↔ ∀x ∈ A x ⊆ B)
4		dfss3 3264	. 2 ⊢ (A ⊆ ℘B ↔ ∀x ∈ A x ∈ ℘B)
5		unissb 3922	. 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 2615   ⊆ wss 3258  ℘cpw 3723  ∪cuni 3892

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 2479  df-ral 2620  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725  df-uni 3893

This theorem is referenced by:  pwssb  4053  elpwuni  4054  rintn0  4057  qsss  5987