Theorem pwss 3737

Description: Subclass relationship for power class. (Contributed by NM, 21-Jun-2009.)

Assertion

Ref	Expression
pwss	⊢ (℘A ⊆ B ↔ ∀x(x ⊆ A → x ∈ B))

Distinct variable groups:   x,A   x,B

Proof of Theorem pwss

Step	Hyp	Ref	Expression
1		dfss2 3263	. 2 ⊢ (℘A ⊆ B ↔ ∀x(x ∈ ℘A → x ∈ B))
2		df-pw 3725	. . . . 5 ⊢ ℘A = {x ∣ x ⊆ A}
3	2	abeq2i 2461	. . . 4 ⊢ (x ∈ ℘A ↔ x ⊆ A)
4	3	imbi1i 315	. . 3 ⊢ ((x ∈ ℘A → x ∈ B) ↔ (x ⊆ A → x ∈ B))
5	4	albii 1566	. 2 ⊢ (∀x(x ∈ ℘A → x ∈ B) ↔ ∀x(x ⊆ A → x ∈ B))
6	1, 5	bitri 240	1 ⊢ (℘A ⊆ B ↔ ∀x(x ⊆ A → x ∈ B))

Syntax hints:   → wi 4   ↔ wb 176  ∀wal 1540   ∈ 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: (None)