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Theorem pwss 4433
 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 3839 . 2 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵))
2 selpw 4423 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32imbi1i 342 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥𝐵))
43albii 1783 . 2 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
51, 4bitri 267 1 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 198  ∀wal 1506   ∈ wcel 2051   ⊆ wss 3822  𝒫 cpw 4416 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743 This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-nfc 2911  df-v 3410  df-in 3829  df-ss 3836  df-pw 4418 This theorem is referenced by:  axpweq  5114  setind2  8969  axgroth5  10042  axgroth6  10046  grumnudlem  40034  ismnuprim  40043
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