Theorem sspwb 4119

Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by SF, 13-Oct-1996.)

Assertion

Ref	Expression
sspwb	⊢ (A ⊆ B ↔ ℘A ⊆ ℘B)

Proof of Theorem sspwb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 3280	. . . . 5 ⊢ (x ⊆ A → (A ⊆ B → x ⊆ B))
2	1	com12 27	. . . 4 ⊢ (A ⊆ B → (x ⊆ A → x ⊆ B))
3		vex 2863	. . . . 5 ⊢ x ∈ V
4	3	elpw 3729	. . . 4 ⊢ (x ∈ ℘A ↔ x ⊆ A)
5	3	elpw 3729	. . . 4 ⊢ (x ∈ ℘B ↔ x ⊆ B)
6	2, 4, 5	3imtr4g 261	. . 3 ⊢ (A ⊆ B → (x ∈ ℘A → x ∈ ℘B))
7	6	ssrdv 3279	. 2 ⊢ (A ⊆ B → ℘A ⊆ ℘B)
8		ssel 3268	. . . 4 ⊢ (℘A ⊆ ℘B → ({x} ∈ ℘A → {x} ∈ ℘B))
9		snex 4112	. . . . . 6 ⊢ {x} ∈ V
10	9	elpw 3729	. . . . 5 ⊢ ({x} ∈ ℘A ↔ {x} ⊆ A)
11	3	snss 3839	. . . . 5 ⊢ (x ∈ A ↔ {x} ⊆ A)
12	10, 11	bitr4i 243	. . . 4 ⊢ ({x} ∈ ℘A ↔ x ∈ A)
13	9	elpw 3729	. . . . 5 ⊢ ({x} ∈ ℘B ↔ {x} ⊆ B)
14	3	snss 3839	. . . . 5 ⊢ (x ∈ B ↔ {x} ⊆ B)
15	13, 14	bitr4i 243	. . . 4 ⊢ ({x} ∈ ℘B ↔ x ∈ B)
16	8, 12, 15	3imtr3g 260	. . 3 ⊢ (℘A ⊆ ℘B → (x ∈ A → x ∈ B))
17	16	ssrdv 3279	. 2 ⊢ (℘A ⊆ ℘B → A ⊆ B)
18	7, 17	impbii 180	1 ⊢ (A ⊆ B ↔ ℘A ⊆ ℘B)

This proof depends on syntax axioms:   ↔ wb 176   ∈ wcel 1710   ⊆ wss 3258  ℘cpw 3723  {csn 3738

This proof depends on 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  ax-nin 4079  ax-sn 4088

This proof 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-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-un 3215  df-dif 3216  df-ss 3260  df-nul 3552  df-pw 3725  df-sn 3742

This theorem is used by:  pw1ss  4170  sfinltfin  4536  ce2le  6234