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Theorem sspwb 5454
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 4611 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
2 ssel 3977 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 → ({𝑥} ∈ 𝒫 𝐴 → {𝑥} ∈ 𝒫 𝐵))
3 vsnex 5434 . . . . . 6 {𝑥} ∈ V
43elpw 4604 . . . . 5 ({𝑥} ∈ 𝒫 𝐴 ↔ {𝑥} ⊆ 𝐴)
5 vex 3484 . . . . . 6 𝑥 ∈ V
65snss 4785 . . . . 5 (𝑥𝐴 ↔ {𝑥} ⊆ 𝐴)
74, 6bitr4i 278 . . . 4 ({𝑥} ∈ 𝒫 𝐴𝑥𝐴)
83elpw 4604 . . . . 5 ({𝑥} ∈ 𝒫 𝐵 ↔ {𝑥} ⊆ 𝐵)
95snss 4785 . . . . 5 (𝑥𝐵 ↔ {𝑥} ⊆ 𝐵)
108, 9bitr4i 278 . . . 4 ({𝑥} ∈ 𝒫 𝐵𝑥𝐵)
112, 7, 103imtr3g 295 . . 3 (𝒫 𝐴 ⊆ 𝒫 𝐵 → (𝑥𝐴𝑥𝐵))
1211ssrdv 3989 . 2 (𝒫 𝐴 ⊆ 𝒫 𝐵𝐴𝐵)
131, 12impbii 209 1 (𝐴𝐵 ↔ 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wb 206  wcel 2108  wss 3951  𝒫 cpw 4600  {csn 4626
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1543  df-ex 1780  df-sb 2065  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3482  df-un 3956  df-ss 3968  df-pw 4602  df-sn 4627  df-pr 4629
This theorem is referenced by:  ssextss  5458  pweqb  5461  psspwb  42267
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