Theorem sspwb 3943

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

Assertion

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

Proof of Theorem sspwb

Dummy variable x is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sstr2 2946	. . . . 5 ⊢ (x ⊆ A → (A ⊆ B → x ⊆ B))
2	1	com12 27	. . . 4 ⊢ (A ⊆ B → (x ⊆ A → x ⊆ B))
3		vex 2554	. . . . 5 ⊢ x ∈ V
4	3	elpw 3357	. . . 4 ⊢ (x ∈ 𝒫 A ↔ x ⊆ A)
5	3	elpw 3357	. . . 4 ⊢ (x ∈ 𝒫 B ↔ x ⊆ B)
6	2, 4, 5	3imtr4g 194	. . 3 ⊢ (A ⊆ B → (x ∈ 𝒫 A → x ∈ 𝒫 B))
7	6	ssrdv 2945	. 2 ⊢ (A ⊆ B → 𝒫 A ⊆ 𝒫 B)
8		ssel 2933	. . . 4 ⊢ (𝒫 A ⊆ 𝒫 B → ({x} ∈ 𝒫 A → {x} ∈ 𝒫 B))
9		snexgOLD 3926	. . . . . . 7 ⊢ (x ∈ V → {x} ∈ V)
10	3, 9	ax-mp 7	. . . . . 6 ⊢ {x} ∈ V
11	10	elpw 3357	. . . . 5 ⊢ ({x} ∈ 𝒫 A ↔ {x} ⊆ A)
12	3	snss 3485	. . . . 5 ⊢ (x ∈ A ↔ {x} ⊆ A)
13	11, 12	bitr4i 176	. . . 4 ⊢ ({x} ∈ 𝒫 A ↔ x ∈ A)
14	10	elpw 3357	. . . . 5 ⊢ ({x} ∈ 𝒫 B ↔ {x} ⊆ B)
15	3	snss 3485	. . . . 5 ⊢ (x ∈ B ↔ {x} ⊆ B)
16	14, 15	bitr4i 176	. . . 4 ⊢ ({x} ∈ 𝒫 B ↔ x ∈ B)
17	8, 13, 16	3imtr3g 193	. . 3 ⊢ (𝒫 A ⊆ 𝒫 B → (x ∈ A → x ∈ B))
18	17	ssrdv 2945	. 2 ⊢ (𝒫 A ⊆ 𝒫 B → A ⊆ B)
19	7, 18	impbii 117	1 ⊢ (A ⊆ B ↔ 𝒫 A ⊆ 𝒫 B)

Syntax hints:   ↔ wb 98   ∈ wcel 1390  Vcvv 2551   ⊆ wss 2911  𝒫 cpw 3351  {csn 3367

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918

This theorem depends on definitions:  df-bi 110  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-v 2553  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373

This theorem is referenced by:  pwel  3945  ssextss  3947  pweqb  3950