Theorem pwssb 5048

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 5047	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2		unissb 4889	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
3	1, 2	bitri 277	1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

Syntax hints:   ↔ wb 208  ∀wral 3066   ⊆ wss 3895  𝒫 cpw 4545  ∪ cuni 4855

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1805  ax-4 1819  ax-5 1920  ax-6 1977  ax-7 2018  ax-8 2134  ax-9 2142  ax-ext 2724

This theorem depends on definitions:  df-bi 209  df-an 399  df-tru 1553  df-ex 1790  df-sb 2081  df-clab 2731  df-cleq 2744  df-clel 2827  df-ral 3067  df-v 3446  df-ss 3912  df-pw 4547  df-uni 4856

This theorem is referenced by:  ustuni  24255  metustfbas  24586  intlidl  33552  dmvlsiga  34370  1stmbfm  34501  2ndmbfm  34502  dya2iocucvr  34525  gneispace  44648  preimafvsspwdm  47933  usgrexmpl1lem  48581  usgrexmpl2lem  48586