Metamath Proof Explorer


Theorem sspwimpALT

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 is the completed proof in conventional notation of the Virtual Deduction proof https://us.metamath.org/other/completeusersproof/sspwimpaltvd.html . It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 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 ). 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 ). (Contributed by Alan Sare, 3-Dec-2015) (Proof modification is discouraged.) (New usage is discouraged.)

Ref Expression
Assertion sspwimpALT A B 𝒫 A 𝒫 B

Proof

Step Hyp Ref Expression
1 vex x V
2 1 a1i x V
3 id x 𝒫 A x 𝒫 A
4 elpwi x 𝒫 A x A
5 3 4 syl x 𝒫 A x A
6 id A B A B
7 5 6 sylan9ssr A B x 𝒫 A x B
8 2 7 elpwgded A B x 𝒫 A x 𝒫 B
9 8 uunT1 A B x 𝒫 A x 𝒫 B
10 9 ex A B x 𝒫 A x 𝒫 B
11 10 alrimiv A B x x 𝒫 A x 𝒫 B
12 dfss2 𝒫 A 𝒫 B x x 𝒫 A x 𝒫 B
13 12 biimpri x x 𝒫 A x 𝒫 B 𝒫 A 𝒫 B
14 11 13 syl A B 𝒫 A 𝒫 B
15 14 idiALT A B 𝒫 A 𝒫 B