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Theorem sspwimpALT2 37969
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 3175 . . . 4 𝑥 ∈ V
2 elpwi 4116 . . . . 5 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
3 id 22 . . . . 5 (𝐴𝐵𝐴𝐵)
42, 3sylan9ssr 3581 . . . 4 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
5 elpwg 4115 . . . . 5 (𝑥 ∈ V → (𝑥 ∈ 𝒫 𝐵𝑥𝐵))
65biimpar 500 . . . 4 ((𝑥 ∈ V ∧ 𝑥𝐵) → 𝑥 ∈ 𝒫 𝐵)
71, 4, 6sylancr 693 . . 3 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
87ex 448 . 2 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
98ssrdv 3573 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wcel 1976  Vcvv 3172  wss 3539  𝒫 cpw 4107
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-v 3174  df-in 3546  df-ss 3553  df-pw 4109
This theorem is referenced by: (None)
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