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Theorem sspwimpALT 28701
 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 28701 is the completed proof in conventional notation of the Virtual Deduction proof http://www.virtualdeduction.com/sspwimpaltvd.html. It was completed manually. The potential for automated derivation from the VD proof exists. See wvd1 28337 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 28330). 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 3633) . (Contributed by Alan Sare, 3-Dec-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimpALT

Proof of Theorem sspwimpALT
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 2791 . . . . . . . 8
21a1i 10 . . . . . . 7
3 id 19 . . . . . . . . 9
4 elpwi 3633 . . . . . . . . 9
53, 4syl 15 . . . . . . . 8
6 id 19 . . . . . . . 8
75, 6sylan9ssr 3193 . . . . . . 7
82, 7elpwgded 28330 . . . . . 6
98uunT1 28555 . . . . 5
109ex 423 . . . 4
1110alrimiv 1617 . . 3
12 dfss2 3169 . . . 4
1312biimpri 197 . . 3
1411, 13syl 15 . 2
1514idi 2 1
 Colors of variables: wff set class Syntax hints:   wi 4   wa 358   wtru 1307  wal 1527   wcel 1684  cvv 2788   wss 3152  cpw 3625 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1635  ax-8 1643  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264 This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1630  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-v 2790  df-in 3159  df-ss 3166  df-pw 3627
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