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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 ↔ ∀𝑥𝐴 𝑥𝐵)
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwssb
StepHypRef Expression
1 sspwuni 5100 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissb 4939 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
31, 2bitri 275 1 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
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
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