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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  |-  ( A 
C_  B  ->  ( A  \  B )  =  (/) )

Proof of Theorem ssdif0im
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 imanim 677 . . . 4  |-  ( ( x  e.  A  ->  x  e.  B )  ->  -.  ( x  e.  A  /\  -.  x  e.  B ) )
2 eldif 3075 . . . 4  |-  ( x  e.  ( A  \  B )  <->  ( x  e.  A  /\  -.  x  e.  B ) )
31, 2sylnibr 666 . . 3  |-  ( ( x  e.  A  ->  x  e.  B )  ->  -.  x  e.  ( A  \  B ) )
43alimi 1431 . 2  |-  ( A. x ( x  e.  A  ->  x  e.  B )  ->  A. x  -.  x  e.  ( A  \  B ) )
5 dfss2 3081 . 2  |-  ( A 
C_  B  <->  A. x
( x  e.  A  ->  x  e.  B ) )
6 eq0 3376 . 2  |-  ( ( A  \  B )  =  (/)  <->  A. x  -.  x  e.  ( A  \  B
) )
74, 5, 63imtr4i 200 1  |-  ( A 
C_  B  ->  ( A  \  B )  =  (/) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103   A.wal 1329    = wceq 1331    e. wcel 1480    \ cdif 3063    C_ 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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