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Theorem pwss 4555
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 3952 . 2 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵))
2 velpw 4543 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32imbi1i 351 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥𝐵))
43albii 1811 . 2 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
51, 4bitri 276 1 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 207  wal 1526  wcel 2105  wss 3933  𝒫 cpw 4535
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902  ax-6 1961  ax-7 2006  ax-8 2107  ax-9 2115  ax-10 2136  ax-11 2151  ax-12 2167  ax-ext 2790
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 842  df-tru 1531  df-ex 1772  df-nf 1776  df-sb 2061  df-clab 2797  df-cleq 2811  df-clel 2890  df-nfc 2960  df-v 3494  df-in 3940  df-ss 3949  df-pw 4537
This theorem is referenced by:  axpweq  5256  setind2  9165  axgroth5  10234  axgroth6  10238  grumnudlem  40498  ismnuprim  40507
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