Theorem pwssb 4053

Description: Two ways to express a collection of subclasses. (Contributed by NM, 19-Jul-2006.)

Assertion

Ref	Expression
pwssb	⊢ (A ⊆ ℘B ↔ ∀x ∈ A x ⊆ B)

Distinct variable groups:   x,A   x,B

Proof of Theorem pwssb

Step	Hyp	Ref	Expression
1		sspwuni 4052	. 2 ⊢ (A ⊆ ℘B ↔ ∪A ⊆ B)
2		unissb 3922	. 2 ⊢ (∪A ⊆ B ↔ ∀x ∈ A x ⊆ B)
3	1, 2	bitri 240	1 ⊢ (A ⊆ ℘B ↔ ∀x ∈ A x ⊆ B)

Syntax hints:   ↔ wb 176  ∀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: (None)