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Theorem pwssb 3769
 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 3768 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissb 3639 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
31, 2bitri 182 1 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 103  ∀wral 2349   ⊆ wss 2974  𝒫 cpw 3390  ∪ cuni 3609 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064 This theorem depends on definitions:  df-bi 115  df-tru 1288  df-nf 1391  df-sb 1687  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ral 2354  df-v 2604  df-in 2980  df-ss 2987  df-pw 3392  df-uni 3610 This theorem is referenced by: (None)
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