Theorem elpw 3680

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 3265	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		df-pw 3676	. 2 ⊢ 𝒫 𝐵 = {𝑥 ∣ 𝑥 ⊆ 𝐵}
4	1, 2, 3	elab2 2968	1 ⊢ (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵)

Syntax hints:   ↔ wb 105   ∈ wcel 2205  Vcvv 2815   ⊆ wss 3214  𝒫 cpw 3674

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2216

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-v 2817  df-in 3220  df-ss 3227  df-pw 3676

This theorem is referenced by:  velpw  3681  elpwg  3682  prsspw  3874  pwprss  3915  pwtpss  3916  pwv  3918  sspwuni  4081  iinpw  4087  iunpwss  4088  0elpw  4282  pwuni  4310  snelpw  4333  sspwb  4337  ssextss  4341  pwin  4408  pwunss  4409  iunpw  4606  xpsspw  4867  ssenen  7118  pw1ne3  7553  3nsssucpw1  7559  ioof  10323  hashfibclem  11231  ballotfilemth  13225  tgdom  15049  distop  15062  epttop  15067  resttopon  15148  txuni2  15233  umgrbien  16217  umgredg  16252