Theorem pwssb 5060

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 5059	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2		unissb 4899	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
3	1, 2	bitri 275	1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

Syntax hints:   ↔ wb 206  ∀wral 3044   ⊆ wss 3911  𝒫 cpw 4559  ∪ cuni 4867

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-v 3446  df-ss 3928  df-pw 4561  df-uni 4868

This theorem is referenced by:  ustuni  24090  metustfbas  24421  intlidl  33364  dmvlsiga  34092  1stmbfm  34224  2ndmbfm  34225  dya2iocucvr  34248  gneispace  44096  preimafvsspwdm  47363  usgrexmpl1lem  47985  usgrexmpl2lem  47990