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Theorem pwssb 5006
 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 5005 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissb 4853 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
31, 2bitri 278 1 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 Colors of variables: wff setvar class Syntax hints:   ↔ wb 209  ∀wral 3132   ⊆ wss 3918  𝒫 cpw 4520  ∪ cuni 4821 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3137  df-v 3481  df-in 3925  df-ss 3935  df-pw 4522  df-uni 4822 This theorem is referenced by:  ustuni  22823  metustfbas  23155  dmvlsiga  31408  1stmbfm  31538  2ndmbfm  31539  dya2iocucvr  31562  gneispace  40687  preimafvsspwdm  43763
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