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Theorem elpwg 3729
 Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g in set.mm. (Contributed by NM, 6-Aug-2000.)
Assertion
Ref Expression
elpwg (A V → (A BA B))

Proof of Theorem elpwg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2413 . 2 (x = A → (x BA B))
2 sseq1 3292 . 2 (x = A → (x BA B))
3 vex 2862 . . 3 x V
43elpw 3728 . 2 (x Bx B)
51, 2, 4vtoclbg 2915 1 (A V → (A BA B))
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ↔ wb 176   ∈ 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:  elpwi  3730  pwidg  3734  elpmg  6013  nenpw1pwlem2  6085
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