Theorem elpw 3583

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 3180	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		df-pw 3579	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
4	1, 2, 3	elab2 2887	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2148  Vcvv 2739   ⊆ wss 3131  𝒫 cpw 3577

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2741  df-in 3137  df-ss 3144  df-pw 3579

This theorem is referenced by:  velpw  3584  elpwg  3585  prsspw  3767  pwprss  3807  pwtpss  3808  pwv  3810  sspwuni  3973  iinpw  3979  iunpwss  3980  0elpw  4166  pwuni  4194  snelpw  4215  sspwb  4218  ssextss  4222  pwin  4284  pwunss  4285  iunpw  4482  xpsspw  4740  ssenen  6853  pw1ne3  7231  3nsssucpw1  7237  ioof  9973  tgdom  13657  distop  13670  epttop  13675  resttopon  13756  txuni2  13841