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Theorem ssdif0im 3286
 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

Proof of Theorem ssdif0im
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 imanim 785 . . . 4
2 eldif 2927 . . . 4
31, 2sylnibr 602 . . 3
43alimi 1344 . 2
5 dfss2 2934 . 2
6 eq0 3239 . 2
74, 5, 63imtr4i 190 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 97  wal 1241   wceq 1243   wcel 1393   cdif 2914   wss 2917  c0 3224 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 2559  df-dif 2920  df-in 2924  df-ss 2931  df-nul 3225 This theorem is referenced by:  vdif0im  3287  difrab0eqim  3288  difid  3292  difin0  3297
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