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Theorem ssdif0im 3422
 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 677 . . . 4
2 eldif 3075 . . . 4
31, 2sylnibr 666 . . 3
43alimi 1431 . 2
5 dfss2 3081 . 2
6 eq0 3376 . 2
74, 5, 63imtr4i 200 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wa 103  wal 1329   wceq 1331   wcel 1480   cdif 3063   wss 3066  c0 3358 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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119 This theorem depends on definitions:  df-bi 116  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-v 2683  df-dif 3068  df-in 3072  df-ss 3079  df-nul 3359 This theorem is referenced by:  vdif0im  3423  difrab0eqim  3424  difid  3426  difin0  3431
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