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Theorem pwss 3736
 Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)
Assertion
Ref Expression
pwss (A Bx(x Ax B))
Distinct variable groups:   x,A   x,B

Proof of Theorem pwss
StepHypRef Expression
1 dfss2 3262 . 2 (A Bx(x Ax B))
2 df-pw 3724 . . . . 5 A = {x x A}
32abeq2i 2460 . . . 4 (x Ax A)
43imbi1i 315 . . 3 ((x Ax B) ↔ (x Ax B))
54albii 1566 . 2 (x(x Ax B) ↔ x(x Ax B))
61, 5bitri 240 1 (A Bx(x Ax B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ wcel 1710   ⊆ wss 3257  ℘cpw 3722 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2478  df-v 2861  df-nin 3211  df-compl 3212  df-in 3213  df-ss 3259  df-pw 3724 This theorem is referenced by: (None)
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