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Theorem sspwimpALT2 38186
 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. Proof derived by completeusersproof.c from User's Proof in VirtualDeductionProofs.txt. The User's Proof in html format is displayed in http://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. (Contributed by Alan Sare, 11-Sep-2016.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpALT2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpALT2
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . 4 𝑥 ∈ V
2 elpwi 4117 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 id 22 . . . . 5 (𝐴𝐵𝐴𝐵)
42, 3sylan9ssr 3582 . . . 4 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
5 elpwg 4116 . . . . 5 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
65biimpar 501 . . . 4 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
71, 4, 6sylancr 694 . . 3 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
87ex 449 . 2 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
98ssrdv 3574 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383   ∈ wcel 1977  Vcvv 3173   ⊆ wss 3540  𝒫 cpw 4108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-in 3547  df-ss 3554  df-pw 4110 This theorem is referenced by: (None)
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