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Theorem nelss 26118
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 3149 . . 3  |-  ( B 
C_  C  ->  ( A  e.  B  ->  A  e.  C ) )
21com12 29 . 2  |-  ( A  e.  B  ->  ( B  C_  C  ->  A  e.  C ) )
32con3and 430 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  C_  C )
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6    /\ wa 360    e. wcel 1621    C_ wss 3127
This theorem is referenced by:  frlmssuvc2  26614
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-5 1533  ax-6 1534  ax-7 1535  ax-gen 1536  ax-8 1623  ax-11 1624  ax-17 1628  ax-12o 1664  ax-10 1678  ax-9 1684  ax-4 1692  ax-16 1927  ax-ext 2239
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1315  df-ex 1538  df-nf 1540  df-sb 1884  df-clab 2245  df-cleq 2251  df-clel 2254  df-in 3134  df-ss 3141
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