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Theorem pwss 4622
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 df-ss 3967 . 2 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵))
2 velpw 4604 . . . 4 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
32imbi1i 349 . . 3 ((𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ (𝑥𝐴𝑥𝐵))
43albii 1818 . 2 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥𝐵) ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
51, 4bitri 275 1 (𝒫 𝐴𝐵 ↔ ∀𝑥(𝑥𝐴𝑥𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wb 206  wal 1537  wcel 2107  wss 3950  𝒫 cpw 4599
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1794  ax-4 1808  ax-5 1909  ax-6 1966  ax-7 2006  ax-8 2109  ax-9 2117  ax-ext 2707
This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1542  df-ex 1779  df-sb 2064  df-clab 2714  df-cleq 2728  df-clel 2815  df-v 3481  df-ss 3967  df-pw 4601
This theorem is referenced by:  axpweq  5350  setind2  9776  axgroth5  10865  axgroth6  10869  grumnudlem  44309  ismnuprim  44318
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