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Theorem sspwimp 41997
 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. For the biconditional, see sspwb 5310. The proof sspwimp 41997, using conventional notation, was translated from virtual deduction form, sspwimpVD 41998, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimp (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)

Proof of Theorem sspwimp
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 vex 3413 . . . . . . 7 𝑥 ∈ V
21a1i 11 . . . . . 6 (⊤ → 𝑥 ∈ V)
3 id 22 . . . . . . 7 (𝐴𝐵𝐴𝐵)
4 id 22 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥 ∈ 𝒫 𝐴)
5 elpwi 4503 . . . . . . . 8 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
64, 5syl 17 . . . . . . 7 (𝑥 ∈ 𝒫 𝐴𝑥𝐴)
7 sstr 3900 . . . . . . . 8 ((𝑥𝐴𝐴𝐵) → 𝑥𝐵)
87ancoms 462 . . . . . . 7 ((𝐴𝐵𝑥𝐴) → 𝑥𝐵)
93, 6, 8syl2an 598 . . . . . 6 ((𝐴𝐵𝑥 ∈ 𝒫 𝐴) → 𝑥𝐵)
102, 9elpwgded 41643 . . . . . 6 ((⊤ ∧ (𝐴𝐵𝑥 ∈ 𝒫 𝐴)) → 𝑥 ∈ 𝒫 𝐵)
112, 9, 10uun0.1 41857 . . . . 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 3878 . . . 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 41651 1 (𝐴𝐵 → 𝒫 𝐴 ⊆ 𝒫 𝐵)
 Colors of variables: wff setvar class Syntax hints:   → wi 4   ∧ wa 399  ∀wal 1536  ⊤wtru 1539   ∈ wcel 2111  Vcvv 3409   ⊆ wss 3858  𝒫 cpw 4494 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 2729 This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1541  df-ex 1782  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-v 3411  df-in 3865  df-ss 3875  df-pw 4496 This theorem is referenced by: (None)
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