Theorem sspwb 5310

Description: The powerclass construction preserves and reflects inclusion. 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		sspw 4507	. 2 ⊢ (𝐴 ⊆ 𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
2		ssel 3885	. . . 4 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
3		snex 5300	. . . . . 6 ⊢ {𝑥} ∈ V
4	3	elpw 4498	. . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
5		vex 3413	. . . . . 6 ⊢ 𝑥 ∈ V
6	5	snss 4676	. . . . 5 ⊢ (𝑥 ∈ 𝐴 ↔ {𝑥} ⊆ 𝐴)
7	4, 6	bitr4i 281	. . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐴 ↔ 𝑥 ∈ 𝐴)
8	3	elpw 4498	. . . . 5 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
9	5	snss 4676	. . . . 5 ⊢ (𝑥 ∈ 𝐵 ↔ {𝑥} ⊆ 𝐵)
10	8, 9	bitr4i 281	. . . 4 ⊢ ({𝑥} ∈ 𝒫 𝐵 ↔ 𝑥 ∈ 𝐵)
11	2, 7, 10	3imtr3g 298	. . 3 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐵))
12	11	ssrdv 3898	. 2 ⊢ (𝒫 𝐴 ⊆ 𝒫 𝐵 → 𝐴 ⊆ 𝐵)
13	1, 12	impbii 212	1 ⊢ (𝐴 ⊆ 𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)

Syntax hints:   ↔ wb 209   ∈ wcel 2111   ⊆ wss 3858  𝒫 cpw 4494  {csn 4522

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pr 5298

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-tru 1541  df-fal 1551  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-pw 4496  df-sn 4523  df-pr 4525

This theorem is referenced by:  ssextss  5314  pweqb  5317  psspwb  39730