Intuitionistic Logic Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  ILE Home  >  Th. List  >  sspwb Unicode version

Theorem sspwb 4176
 Description: Classes are subclasses if and only if their power classes are subclasses. Exercise 18 of [TakeutiZaring] p. 18. (Contributed by NM, 13-Oct-1996.)
Assertion
Ref Expression
sspwb

Proof of Theorem sspwb
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sstr2 3135 . . . . 5
21com12 30 . . . 4
3 vex 2715 . . . . 5
43elpw 3549 . . . 4
53elpw 3549 . . . 4
62, 4, 53imtr4g 204 . . 3
76ssrdv 3134 . 2
8 ssel 3122 . . . 4
93snex 4146 . . . . . 6
109elpw 3549 . . . . 5
113snss 3685 . . . . 5
1210, 11bitr4i 186 . . . 4
139elpw 3549 . . . . 5
143snss 3685 . . . . 5
1513, 14bitr4i 186 . . . 4
168, 12, 153imtr3g 203 . . 3
1716ssrdv 3134 . 2
187, 17impbii 125 1
 Colors of variables: wff set class Syntax hints:   wb 104   wcel 2128   wss 3102  cpw 3543  csn 3560 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-pow 4135 This theorem depends on definitions:  df-bi 116  df-tru 1338  df-nf 1441  df-sb 1743  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-v 2714  df-in 3108  df-ss 3115  df-pw 3545  df-sn 3566 This theorem is referenced by:  pwel  4178  ssextss  4180  pweqb  4183  fiss  6918  pw1on  7155  ntrss  12490
 Copyright terms: Public domain W3C validator