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Theorem pwssb 4542
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 4541 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissb 4399 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
31, 2bitri 262 1 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 194  wral 2895  wss 3539  𝒫 cpw 4107   cuni 4366
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2032  ax-13 2232  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-v 3174  df-in 3546  df-ss 3553  df-pw 4109  df-uni 4367
This theorem is referenced by:  ustuni  21782  metustfbas  22113  dmvlsiga  29325  1stmbfm  29455  2ndmbfm  29456  dya2iocucvr  29479  gneispace  37255
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