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Theorem difneqnul 27617
Description: If the difference of two sets is not empty, then the sets are not equal. (Contributed by Thierry Arnoux, 28-Feb-2017.)
Assertion
Ref Expression
difneqnul  |-  ( ( A  \  B )  =/=  (/)  ->  A  =/=  B )

Proof of Theorem difneqnul
StepHypRef Expression
1 eqimss 3541 . . 3  |-  ( A  =  B  ->  A  C_  B )
2 ssdif0 3873 . . 3  |-  ( A 
C_  B  <->  ( A  \  B )  =  (/) )
31, 2sylib 196 . 2  |-  ( A  =  B  ->  ( A  \  B )  =  (/) )
43necon3i 2694 1  |-  ( ( A  \  B )  =/=  (/)  ->  A  =/=  B )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    =/= wne 2649    \ cdif 3458    C_ wss 3461   (/)c0 3783
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-v 3108  df-dif 3464  df-in 3468  df-ss 3475  df-nul 3784
This theorem is referenced by:  disjdsct  27752  bj-2upln1upl  35002
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