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Theorem pwss 3559
Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
Assertion
Ref Expression
pwss (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Distinct variable groups:   𝑥,𝐴   𝑥,𝐵

Proof of Theorem pwss
StepHypRef Expression
1 dfss2 3117 . 2 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵))
2 df-pw 3545 . . . . 5 𝒫 𝐴 = {𝑥𝑥𝐴}
32abeq2i 2268 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
43imbi1i 237 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥𝐵))
54albii 1450 . 2 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
61, 5bitri 183 1 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wal 1333  wcel 2128  wss 3102  𝒫 cpw 3543
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-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-11 1486  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2139
This theorem depends on definitions:  df-bi 116  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-in 3108  df-ss 3115  df-pw 3545
This theorem is referenced by:  axpweq  4132  setind2  4499
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