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Theorem sspwimpcf 39811
Description: If a class is a subclass of another class, then its power class is a subclass of that other class's power class. Left-to-right implication of Exercise 18 of [TakeutiZaring] p. 18. sspwimpcf 39811, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 39812, using a translation program. (Contributed by Alan Sare, 13-Jun-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpcf (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpcf
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3353 . . . . . 6 𝑥 ∈ V
2 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
3 id 22 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐴)
4 elpwi 4327 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53, 4syl 17 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
6 sstr2 3770 . . . . . . . 8 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
76impcom 396 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
82, 5, 7syl2an 589 . . . . . 6 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
9 elpwg 4325 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
109biimpar 469 . . . . . 6 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
111, 8, 10eel021old 39590 . . . . 5 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
1211ex 401 . . . 4 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1312alrimiv 2022 . . 3 (𝐴𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
14 dfss2 3751 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 219 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15syl 17 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1716iin1 39453 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wal 1650  wcel 2155  Vcvv 3350  wss 3734  𝒫 cpw 4317
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1890  ax-4 1904  ax-5 2005  ax-6 2070  ax-7 2105  ax-9 2164  ax-10 2183  ax-11 2198  ax-12 2211  ax-ext 2743
This theorem depends on definitions:  df-bi 198  df-an 385  df-or 874  df-tru 1656  df-ex 1875  df-nf 1879  df-sb 2063  df-clab 2752  df-cleq 2758  df-clel 2761  df-nfc 2896  df-v 3352  df-in 3741  df-ss 3748  df-pw 4319
This theorem is referenced by: (None)
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