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Theorem elpw2g 3937
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 3395 . 2 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
2 ssexg 3923 . . . 4 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ V)
3 elpwg 3394 . . . . 5 (𝐴 ∈ V → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
43biimparc 287 . . . 4 ((𝐴𝐵𝐴 ∈ V) → 𝐴 ∈ 𝒫 𝐵)
52, 4syldan 270 . . 3 ((𝐴𝐵𝐵𝑉) → 𝐴 ∈ 𝒫 𝐵)
65expcom 113 . 2 (𝐵𝑉 → (𝐴𝐵𝐴 ∈ 𝒫 𝐵))
71, 6impbid2 135 1 (𝐵𝑉 → (𝐴 ∈ 𝒫 𝐵𝐴𝐵))
Colors of variables: wff set class
Syntax hints:  wi 4  wb 102  wcel 1409  Vcvv 2574  wss 2944  𝒫 cpw 3386
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2951  df-ss 2958  df-pw 3388
This theorem is referenced by:  elpw2  3938  pwnss  3939
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