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Theorem elpwg 3359
 Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)
Assertion
Ref Expression
elpwg (A 𝑉 → (A 𝒫 BAB))

Proof of Theorem elpwg
Dummy variable x is distinct from all other variables.
StepHypRef Expression
1 eleq1 2097 . 2 (x = A → (x 𝒫 BA 𝒫 B))
2 sseq1 2960 . 2 (x = A → (xBAB))
3 vex 2554 . . 3 x V
43elpw 3357 . 2 (x 𝒫 BxB)
51, 2, 4vtoclbg 2608 1 (A 𝑉 → (A 𝒫 BAB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390   ⊆ wss 2911  𝒫 cpw 3351 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019 This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353 This theorem is referenced by:  elpwi  3360  pwidg  3364  prsspwg  3514  elpw2g  3901  snelpwi  3939  prelpwi  3941  pwel  3945  eldifpw  4174  f1opw2  5648  2pwuninelg  5839  tfrlemibfn  5883  fopwdom  6246
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