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Theorem nelss 3476
Description: Demonstrate by witnesses that two classes lack a subclass relation. (Contributed by Stefan O'Rear, 5-Feb-2015.)
Assertion
Ref Expression
nelss  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  C_  C )

Proof of Theorem nelss
StepHypRef Expression
1 ssel 3411 . . 3  |-  ( B 
C_  C  ->  ( A  e.  B  ->  A  e.  C ) )
21com12 31 . 2  |-  ( A  e.  B  ->  ( B  C_  C  ->  A  e.  C ) )
32con3dimp 439 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  C_  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 367    e. wcel 1826    C_ wss 3389
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1626  ax-4 1639  ax-5 1712  ax-6 1755  ax-7 1798  ax-10 1845  ax-11 1850  ax-12 1862  ax-13 2006  ax-ext 2360
This theorem depends on definitions:  df-bi 185  df-an 369  df-tru 1402  df-ex 1621  df-nf 1625  df-sb 1748  df-clab 2368  df-cleq 2374  df-clel 2377  df-in 3396  df-ss 3403
This theorem is referenced by:  frlmssuvc2  18915  fourierdlem10  32065
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