Theorem sspwb 1863

Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18.

Assertion

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

Proof of Theorem sspwb

Step	Hyp	Ref	Expression
1		sstr2 1510	. . . . 5 ⊢ (x ⊆ A → (A ⊆ B → x ⊆ B))
2	1	com12 13	. . . 4 ⊢ (A ⊆ B → (x ⊆ A → x ⊆ B))
3		visset 1350	. . . . 5 ⊢ x ∈ V
4	3	elpw 1801	. . . 4 ⊢ (x ∈ ℘A ↔ x ⊆ A)
5	3	elpw 1801	. . . 4 ⊢ (x ∈ ℘B ↔ x ⊆ B)
6	2, 4, 5	3imtr4g 426	. . 3 ⊢ (A ⊆ B → (x ∈ ℘A → x ∈ ℘B))
7	6	ssrdv 1509	. 2 ⊢ (A ⊆ B → ℘A ⊆ ℘B)
8		ssel 1502	. . . 4 ⊢ (℘A ⊆ ℘B → ({x} ∈ ℘A → {x} ∈ ℘B))
9		snex 1859	. . . . . 6 ⊢ {x} ∈ V
10	9	elpw 1801	. . . . 5 ⊢ ({x} ∈ ℘A ↔ {x} ⊆ A)
11	3	snss 1849	. . . . 5 ⊢ (x ∈ A ↔ {x} ⊆ A)
12	10, 11	bitr4 154	. . . 4 ⊢ ({x} ∈ ℘A ↔ x ∈ A)
13	9	elpw 1801	. . . . 5 ⊢ ({x} ∈ ℘B ↔ {x} ⊆ B)
14	3	snss 1849	. . . . 5 ⊢ (x ∈ B ↔ {x} ⊆ B)
15	13, 14	bitr4 154	. . . 4 ⊢ ({x} ∈ ℘B ↔ x ∈ B)
16	8, 12, 15	3imtr3g 425	. . 3 ⊢ (℘A ⊆ ℘B → (x ∈ A → x ∈ B))
17	16	ssrdv 1509	. 2 ⊢ (℘A ⊆ ℘B → A ⊆ B)
18	7, 17	impbi 139	1 ⊢ (A ⊆ B ↔ ℘A ⊆ ℘B)

Syntax hints:   ↔ wb 127   ∈ wcel 1092   ⊆ wss 1487  ℘cpw 1798  {csn 1808

This theorem is referenced by:  ssextss 1864  pweqb 1867  rankpw 3528

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811

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