Theorem elpwg 3730

Description: Membership in a power class. Theorem 86 of [Suppes] p. 47. See also elpw2g in set.mm. (Contributed by NM, 6-Aug-2000.)

Assertion

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

Proof of Theorem elpwg

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		eleq1 2413	. 2 ⊢ (x = A → (x ∈ ℘B ↔ A ∈ ℘B))
2		sseq1 3293	. 2 ⊢ (x = A → (x ⊆ B ↔ A ⊆ B))
3		vex 2863	. . 3 ⊢ x ∈ V
4	3	elpw 3729	. 2 ⊢ (x ∈ ℘B ↔ x ⊆ B)
5	1, 2, 4	vtoclbg 2916	1 ⊢ (A ∈ V → (A ∈ ℘B ↔ A ⊆ B))

Syntax hints:   → wi 4   ↔ wb 176   ∈ wcel 1710   ⊆ wss 3258  ℘cpw 3723

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260  df-pw 3725

This theorem is referenced by:  elpwi  3731  pwidg  3735  elpmg  6014  nenpw1pwlem2  6086