ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ssdif0im Unicode version

Theorem ssdif0im 3283
Description: Subclass implies empty difference. One direction of Exercise 7 of [TakeutiZaring] p. 22. In classical logic this would be an equivalence. (Contributed by Jim Kingdon, 2-Aug-2018.)
Assertion
Ref Expression
ssdif0im 
C_  \  (/)

Proof of Theorem ssdif0im
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 imanim 785 . . . 4
2 eldif 2924 . . . 4  \
31, 2sylnibr 602 . . 3  \
43alimi 1344 . 2  \
5 dfss2 2931 . 2 
C_
6 eq0 3236 . 2  \  (/)  \
74, 5, 63imtr4i 190 1 
C_  \  (/)
Colors of variables: wff set class
Syntax hints:   wn 3   wi 4   wa 97  wal 1241   wceq 1243   wcel 1393    \ cdif 2911    C_ wss 2914   (/)c0 3221
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-in1 544  ax-in2 545  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-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022
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 2556  df-dif 2917  df-in 2921  df-ss 2928  df-nul 3222
This theorem is referenced by:  vdif0im  3284  difrab0eqim  3285  difid  3289  difin0  3294
  Copyright terms: Public domain W3C validator