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Theorem pwssb 3907
 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 3906 . 2 (𝐴 ⊆ 𝒫 𝐵 𝐴𝐵)
2 unissb 3775 . 2 ( 𝐴𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
31, 2bitri 183 1 (𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥𝐴 𝑥𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 104  ∀wral 2417   ⊆ wss 3077  𝒫 cpw 3516  ∪ cuni 3745 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122 This theorem depends on definitions:  df-bi 116  df-tru 1335  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-v 2692  df-in 3083  df-ss 3090  df-pw 3518  df-uni 3746 This theorem is referenced by: (None)
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