Theorem pwssb 5111

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

Assertion

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

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwssb

Step	Hyp	Ref	Expression
1		sspwuni 5110	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2		unissb 4949	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
3	1, 2	bitri 274	1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

Syntax hints:   ↔ wb 205  ∀wral 3051   ⊆ wss 3947  𝒫 cpw 4607  ∪ cuni 4915

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2697

This theorem depends on definitions:  df-bi 206  df-an 395  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2704  df-cleq 2718  df-clel 2803  df-ral 3052  df-v 3464  df-ss 3964  df-pw 4609  df-uni 4916

This theorem is referenced by:  ustuni  24225  metustfbas  24560  intlidl  33297  dmvlsiga  33964  1stmbfm  34096  2ndmbfm  34097  dya2iocucvr  34120  gneispace  43819  preimafvsspwdm  46979