Theorem elpwg 3679

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. (Contributed by NM, 6-Aug-2000.)

Assertion

Ref	Expression
elpwg	⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Proof of Theorem elpwg

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2297	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ∈ 𝒫 𝐵 ↔ 𝐴 ∈ 𝒫 𝐵))
2		sseq1 3263	. 2 ⊢ (𝑥 = 𝐴 → (𝑥 ⊆ 𝐵 ↔ 𝐴 ⊆ 𝐵))
3		vex 2818	. . 3 ⊢ 𝑥 ∈ V
4	3	elpw 3677	. 2 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
5	1, 2, 4	vtoclbg 2878	1 ⊢ (𝐴 ∈ 𝑉 → (𝐴 ∈ 𝒫 𝐵 ↔ 𝐴 ⊆ 𝐵))

Syntax hints:   → wi 4   ↔ wb 105   ∈ wcel 2205   ⊆ wss 3213  𝒫 cpw 3671

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 3219  df-ss 3226  df-pw 3673

This theorem is referenced by:  elpwi  3680  elpwb  3681  pwidg  3688  prsspwg  3856  elpw2g  4270  snelpwg  4328  snelpwi  4329  prelpw  4331  prelpwi  4332  pwel  4336  eldifpw  4600  f1opw2  6263  2pwuninelg  6516  tfrlemibfn  6561  tfr1onlembfn  6577  tfrcllembfn  6590  elpmg  6900  pw2f1odclem  7089  fopwdom  7091  elfpw  7217  fiinopn  14918  ssntr  15036  incistruhgr  16134  upgr1edc  16165  uspgr1edc  16284  uhgrspansubgrlem  16320  eupth2lemsfi  16522