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Theorem sspwimpALT 38183
 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. sspwimpALT 38183 is the completed proof in conventional notation of the Virtual Deduction proof http://us.metamath.org/other/completeusersproof/sspwimpaltvd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 37806 for a description of Virtual Deduction. Some sub-theorems of the proof were completed using a unification deduction (e.g., the sub-theorem whose assertion is step 9 used elpwgded 37801). Unification deductions employ Mario Carneiro's metavariable concept. Some sub-theorems were completed using a unification theorem (e.g., the sub-theorem whose assertion is step 5 used elpwi 4117). (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpALT (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimpALT
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3176 . . . . . . . 8 𝑥 ∈ V
21a1i 11 . . . . . . 7 (⊤ → 𝑥 ∈ V)
3 id 22 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐴)
4 elpwi 4117 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53, 4syl 17 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
6 id 22 . . . . . . . 8 (𝐴𝐵𝐴𝐵)
75, 6sylan9ssr 3582 . . . . . . 7 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
82, 7elpwgded 37801 . . . . . 6 ((⊤ ∧ (𝐴𝐵𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵)
98uunT1 38028 . . . . 5 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
109ex 449 . . . 4 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1110alrimiv 1842 . . 3 (𝐴𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
12 dfss2 3557 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1312biimpri 217 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1411, 13syl 17 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1514idiALT 37704 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 383  ∀wal 1473  ⊤wtru 1476   ∈ 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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