Theorem sspwb 3971

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 3006	. . . . 5 ⊢ (𝑥 ⊆ 𝐴 → (𝐴 ⊆ 𝐵 → 𝑥 ⊆ 𝐵))
2	1	com12 30	. . . 4 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ⊆ 𝐴 → 𝑥 ⊆ 𝐵))
3		vex 2604	. . . . 5 ⊢ 𝑥 ∈ V
4	3	elpw 3388	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐴 ↔ 𝑥 ⊆ 𝐴)
5	3	elpw 3388	. . . 4 ⊢ (𝑥 ∈ 𝒫 𝐵 ↔ 𝑥 ⊆ 𝐵)
6	2, 4, 5	3imtr4g 203	. . 3 ⊢ (𝐴 ⊆ 𝐵 → (𝑥 ∈ 𝒫 𝐴 → 𝑥 ∈ 𝒫 𝐵))
7	6	ssrdv 3005	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
8		ssel 2993	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
9	3	snex 3957	. . . . . 6 ⊢ {𝑥} ∈ V
10	9	elpw 3388	. . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
11	3	snss 3516	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
12	10, 11	bitr4i 185	. . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴)
13	9	elpw 3388	. . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
14	3	snss 3516	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵)
15	13, 14	bitr4i 185	. . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝐵)
16	8, 12, 15	3imtr3g 202	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
17	16	ssrdv 3005	. 2 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
18	7, 17	impbii 124	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   ↔ wb 103   ∈ wcel 1433   ⊆ wss 2973  𝒫 cpw 3382  {csn 3398

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948

This theorem depends on definitions:  df-bi 115  df-tru 1287  df-nf 1390  df-sb 1686  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-v 2603  df-in 2979  df-ss 2986  df-pw 3384  df-sn 3404

This theorem is referenced by:  pwel  3973  ssextss  3975  pweqb  3978