Theorem pwssb 5069

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 5068	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2		unissb 4908	. 2 ⊢ (∪ 𝐴 ⊆ 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)
3	1, 2	bitri 278	1 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥 ∈ 𝐴 𝑥 ⊆ 𝐵)

This proof depends on syntax axioms:   ↔ wb 209  ∀wral 3081   ⊆ wss 3906  𝒫 cpw 4564  ∪ cuni 4874

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2737

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-ex 1813  df-sb 2100  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3082  df-v 3459  df-ss 3923  df-pw 4566  df-uni 4875

This theorem is used by:  ustuni  24439  metustfbas  24770  intlidl  33797  dmvlsiga  34588  1stmbfm  34720  2ndmbfm  34721  dya2iocucvr  34744  gneispace  44938  preimafvsspwdm  48216  usgrexmpl1lem  48864  usgrexmpl2lem  48869