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Theorem sspwb 5451
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 4614 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 ssel 3973 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
3 vsnex 5431 . . . . . 6 {𝑥} ∈ V
43elpw 4607 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
5 vex 3475 . . . . . 6 𝑥 ∈ V
65snss 4790 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
74, 6bitr4i 278 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
83elpw 4607 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
95snss 4790 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
108, 9bitr4i 278 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
112, 7, 103imtr3g 295 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1211ssrdv 3986 . 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 2099  wss 3947  𝒫 cpw 4603  {csn 4629
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5299  ax-pr 5429
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-tru 1537  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-v 3473  df-un 3952  df-in 3954  df-ss 3964  df-pw 4605  df-sn 4630  df-pr 4632
This theorem is referenced by:  ssextss  5455  pweqb  5458  psspwb  41716
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