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Theorem elpw2g 4041
Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 7-Aug-2000.)
Assertion
Ref Expression
elpw2g (𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))

Proof of Theorem elpw2g
StepHypRef Expression
1 elpwi 3485 . 2 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2 ssexg 4027 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
3 elpwg 3484 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimparc 295 . . . 4 ((𝐴𝐵𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵)
52, 4syldan 278 . . 3 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ 𝒫 𝐵)
65expcom 115 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ 𝒫 𝐵))
71, 6impbid2 142 1 (𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 104  wcel 1463  Vcvv 2657  wss 3037  𝒫 cpw 3476
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-sep 4006
This theorem depends on definitions:  df-bi 116  df-tru 1317  df-nf 1420  df-sb 1719  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2244  df-v 2659  df-in 3043  df-ss 3050  df-pw 3478
This theorem is referenced by:  elpw2  4042  pwnss  4043  elfir  6813  istopg  12009  uniopn  12011  iscld  12115  ntrval  12122  clsval  12123  discld  12148  neival  12155  isnei  12156  restdis  12196  cnpfval  12207  cndis  12252  blfvalps  12374  blfps  12398  blf  12399  reldvg  12603
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