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

Theorem sspwb 3952
 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 2952 . . . . 5
21com12 27 . . . 4
3 vex 2560 . . . . 5
43elpw 3365 . . . 4
53elpw 3365 . . . 4
62, 4, 53imtr4g 194 . . 3
76ssrdv 2951 . 2
8 ssel 2939 . . . 4
9 snexgOLD 3935 . . . . . . 7
103, 9ax-mp 7 . . . . . 6
1110elpw 3365 . . . . 5
123snss 3494 . . . . 5
1311, 12bitr4i 176 . . . 4
1410elpw 3365 . . . . 5
153snss 3494 . . . . 5
1614, 15bitr4i 176 . . . 4
178, 13, 163imtr3g 193 . . 3
1817ssrdv 2951 . 2
197, 18impbii 117 1
 Colors of variables: wff set class Syntax hints:   wb 98   wcel 1393  cvv 2557   wss 2917  cpw 3359  csn 3375 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927 This theorem depends on definitions:  df-bi 110  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-v 2559  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381 This theorem is referenced by:  pwel  3954  ssextss  3956  pweqb  3959
 Copyright terms: Public domain W3C validator