Theorem pwssb 5101

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 5100	. 2 ⊢ (𝐴 ⊆ 𝒫 𝐵 ↔ ∪ 𝐴 ⊆ 𝐵)
2		unissb 4939	. 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 3061   ⊆ wss 3951  𝒫 cpw 4600  ∪ cuni 4907

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-v 3482  df-ss 3968  df-pw 4602  df-uni 4908

This theorem is referenced by:  ustuni  24235  metustfbas  24570  intlidl  33448  dmvlsiga  34130  1stmbfm  34262  2ndmbfm  34263  dya2iocucvr  34286  gneispace  44147  preimafvsspwdm  47376  usgrexmpl1lem  47980  usgrexmpl2lem  47985