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Theorem nelss 26854
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 3187 . . 3  |-  ( B 
C_  C  ->  ( A  e.  B  ->  A  e.  C ) )
21com12 27 . 2  |-  ( A  e.  B  ->  ( B  C_  C  ->  A  e.  C ) )
32con3and 428 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  C_  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 358    e. wcel 1696    C_ wss 3165
This theorem is referenced by:  frlmssuvc2  27350
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277
This theorem depends on definitions:  df-bi 177  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-in 3172  df-ss 3179
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