Theorem elpw 3565

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 3165	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		df-pw 3561	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
4	1, 2, 3	elab2 2874	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 104   ∈ wcel 2136  Vcvv 2726   ⊆ wss 3116  𝒫 cpw 3559

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 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-tru 1346  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-v 2728  df-in 3122  df-ss 3129  df-pw 3561

This theorem is referenced by:  velpw  3566  elpwg  3567  prsspw  3745  pwprss  3785  pwtpss  3786  pwv  3788  sspwuni  3950  iinpw  3956  iunpwss  3957  0elpw  4143  pwuni  4171  snelpw  4191  sspwb  4194  ssextss  4198  pwin  4260  pwunss  4261  iunpw  4458  xpsspw  4716  ssenen  6817  pw1ne3  7186  3nsssucpw1  7192  ioof  9907  tgdom  12712  distop  12725  epttop  12730  resttopon  12811  txuni2  12896