Theorem elpw 3511

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 𝐵 ↔ 𝐴 ⊆ 𝐵)

Proof of Theorem elpw

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		elpw.1	. 2 ⊢ 𝐴 ∈ V
2		sseq1 3115	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		df-pw 3507	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
4	1, 2, 3	elab2 2827	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 104   ∈ wcel 1480  Vcvv 2681   ⊆ wss 3066  𝒫 cpw 3505

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119

This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-in 3072  df-ss 3079  df-pw 3507

This theorem is referenced by:  velpw  3512  elpwg  3513  prsspw  3687  pwprss  3727  pwtpss  3728  pwv  3730  sspwuni  3892  iinpw  3898  iunpwss  3899  0elpw  4083  pwuni  4111  snelpw  4130  sspwb  4133  ssextss  4137  pwin  4199  pwunss  4200  iunpw  4396  xpsspw  4646  ssenen  6738  ioof  9747  tgdom  12230  distop  12243  epttop  12248  resttopon  12329  txuni2  12414