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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 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwb
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sspw 4507 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 ssel 3885 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
3 snex 5300 . . . . . 6 {𝑥} ∈ V
43elpw 4498 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
5 vex 3413 . . . . . 6 𝑥 ∈ V
65snss 4676 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
74, 6bitr4i 281 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
83elpw 4498 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
95snss 4676 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
108, 9bitr4i 281 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
112, 7, 103imtr3g 298 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1211ssrdv 3898 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
131, 12impbii 212 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class 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
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