MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  sspwb Structured version   Visualization version   GIF version

Theorem sspwb 5440
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 4606 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 ssel 3968 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
3 vsnex 5420 . . . . . 6 {𝑥} ∈ V
43elpw 4599 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
5 vex 3470 . . . . . 6 𝑥 ∈ V
65snss 4782 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
74, 6bitr4i 278 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
83elpw 4599 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
95snss 4782 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
108, 9bitr4i 278 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
112, 7, 103imtr3g 295 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1211ssrdv 3981 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
131, 12impbii 208 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 205  wcel 2098  wss 3941  𝒫 cpw 4595  {csn 4621
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-tru 1536  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-v 3468  df-un 3946  df-in 3948  df-ss 3958  df-pw 4597  df-sn 4622  df-pr 4624
This theorem is referenced by:  ssextss  5444  pweqb  5447  psspwb  41579
  Copyright terms: Public domain W3C validator