Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  elpw GIF version

Theorem elpw 3365
 Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 31-Dec-1993.)
Hypothesis
Ref Expression
elpw.1 𝐴 ∈ V
Assertion
Ref Expression
elpw (𝐴 ∈ 𝒫 𝐵𝐴𝐵)

Proof of Theorem elpw
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 elpw.1 . 2 𝐴 ∈ V
2 sseq1 2966 . 2 (𝑥 = 𝐴 → (𝑥𝐵𝐴𝐵))
3 df-pw 3361 . 2 𝒫 𝐵 = {𝑥𝑥𝐵}
41, 2, 3elab2 2690 1 (𝐴 ∈ 𝒫 𝐵𝐴𝐵)
 Colors of variables: wff set class Syntax hints:   ↔ wb 98   ∈ wcel 1393  Vcvv 2557   ⊆ wss 2917  𝒫 cpw 3359 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 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361 This theorem is referenced by:  selpw  3366  elpwg  3367  prsspw  3536  pwprss  3576  pwtpss  3577  pwv  3579  sspwuni  3739  iinpw  3742  iunpwss  3743  0elpw  3917  pwuni  3943  snelpw  3949  sspwb  3952  ssextss  3956  pwin  4019  pwunss  4020  iunpw  4211  xpsspw  4450  ioof  8840
 Copyright terms: Public domain W3C validator