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Theorem elpw2g 3901
 Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
Assertion
Ref Expression
elpw2g (B 𝑉 → (A 𝒫 BAB))

Proof of Theorem elpw2g
StepHypRef Expression
1 elpwi 3360 . 2 (A 𝒫 BAB)
2 ssexg 3887 . . . 4 ((AB B 𝑉) → A V)
3 elpwg 3359 . . . . 5 (A V → (A 𝒫 BAB))
43biimparc 283 . . . 4 ((AB A V) → A 𝒫 B)
52, 4syldan 266 . . 3 ((AB B 𝑉) → A 𝒫 B)
65expcom 109 . 2 (B 𝑉 → (ABA 𝒫 B))
71, 6impbid2 131 1 (B 𝑉 → (A 𝒫 BAB))
 Colors of variables: wff set class Syntax hints:   → wi 4   ↔ wb 98   ∈ wcel 1390  Vcvv 2551   ⊆ 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-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866 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:  elpw2  3902  pwnss  3903
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