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Theorem sspwimpALT 39683
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 39683 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 39310 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 39305). 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 4307). (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 3354 . . . . . . . 8 𝑥 ∈ V
21a1i 11 . . . . . . 7 (⊤ → 𝑥 ∈ V)
3 id 22 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐴)
4 elpwi 4307 . . . . . . . . 9 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
53, 4syl 17 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
6 id 22 . . . . . . . 8 (𝐴𝐵𝐴𝐵)
75, 6sylan9ssr 3766 . . . . . . 7 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
82, 7elpwgded 39305 . . . . . 6 ((⊤ ∧ (𝐴𝐵𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵)
98uunT1 39532 . . . . 5 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥 ∈ 𝒫 𝐵)
109ex 397 . . . 4 (𝐴𝐵 → (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1110alrimiv 2007 . . 3 (𝐴𝐵 → ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
12 dfss2 3740 . . . 4 (𝒫 𝐴 ⊆ 𝒫 𝐵 ↔ ∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵))
1312biimpri 218 . . 3 (∀𝑥(𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐵) → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1411, 13syl 17 . 2 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
1514idiALT 39208 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 382  wal 1629  wtru 1632  wcel 2145  Vcvv 3351  wss 3723  𝒫 cpw 4297
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 835  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-v 3353  df-in 3730  df-ss 3737  df-pw 4299
This theorem is referenced by: (None)
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