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Theorem sspwimpcf 41797
 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 41797, using conventional notation, was translated from its virtual deduction form, sspwimpcfVD 41798, 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 3445 . . . . . 6 𝑥 ∈ V
2 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
3 id 22 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐴)
4 elpwi 4509 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53, 4syl 17 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
6 sstr2 3924 . . . . . . . 8 (𝑥𝐴 → (𝐴𝐵𝑥𝐵))
76impcom 411 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
82, 5, 7syl2an 598 . . . . . 6 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
9 elpwg 4503 . . . . . . 7 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
109biimpar 481 . . . . . 6 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
111, 8, 10eel021old 41577 . . . . 5 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
1211ex 416 . . . 4 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1312alrimiv 1928 . . 3 (𝐴𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
14 dfss2 3903 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1514biimpri 231 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1613, 15syl 17 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1716iin1 41449 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4  ∀wal 1536   ∈ wcel 2111  Vcvv 3442   ⊆ wss 3883  𝒫 cpw 4500 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 2770 This theorem depends on definitions:  df-bi 210  df-an 400  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3444  df-in 3890  df-ss 3900  df-pw 4502 This theorem is referenced by: (None)
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