Mathbox for Alan Sare < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  sspwimp Structured version   Visualization version   Unicode version

Theorem sspwimp 37315
 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. sspwimp 37315, using conventional notation, was translated from virtual deduction form, sspwimpVD 37316, using a translation program. (Contributed by Alan Sare, 23-Apr-2015.) (Proof modification is discouraged.) (New usage is discouraged.)
Assertion
Ref Expression
sspwimp

Proof of Theorem sspwimp
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 vex 3048 . . . . . . 7
21a1i 11 . . . . . 6
3 id 22 . . . . . . 7
4 id 22 . . . . . . . 8
5 elpwi 3960 . . . . . . . 8
64, 5syl 17 . . . . . . 7
7 sstr 3440 . . . . . . . 8
87ancoms 455 . . . . . . 7
93, 6, 8syl2an 480 . . . . . 6
102, 9elpwgded 36931 . . . . . 6
112, 9, 10uun0.1 37165 . . . . 5
1211ex 436 . . . 4
1312alrimiv 1773 . . 3
14 dfss2 3421 . . . 4
1514biimpri 210 . . 3
1613, 15syl 17 . 2
1716iin1 36942 1
 Colors of variables: wff setvar class Syntax hints:   wi 4   wa 371  wal 1442   wtru 1445   wcel 1887  cvv 3045   wss 3404  cpw 3951 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1669  ax-4 1682  ax-5 1758  ax-6 1805  ax-7 1851  ax-10 1915  ax-11 1920  ax-12 1933  ax-13 2091  ax-ext 2431 This theorem depends on definitions:  df-bi 189  df-an 373  df-tru 1447  df-ex 1664  df-nf 1668  df-sb 1798  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-v 3047  df-in 3411  df-ss 3418  df-pw 3953 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator