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Theorem pwss 3440
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 3012 . 2 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵))
2 df-pw 3427 . . . . 5 𝒫 𝐴 = {𝑥𝑥𝐴}
32abeq2i 2198 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
43imbi1i 236 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥𝐵))
54albii 1404 . 2 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
61, 5bitri 182 1 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103  wal 1287  wcel 1438  wss 2997  𝒫 cpw 3425
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-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-11 1442  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-in 3003  df-ss 3010  df-pw 3427
This theorem is referenced by:  axpweq  3998  setind2  4346
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