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Theorem nelss 3518
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 3453 . . 3  |-  ( B 
C_  C  ->  ( A  e.  B  ->  A  e.  C ) )
21com12 31 . 2  |-  ( A  e.  B  ->  ( B  C_  C  ->  A  e.  C ) )
32con3dimp 441 1  |-  ( ( A  e.  B  /\  -.  A  e.  C
)  ->  -.  B  C_  C )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    e. wcel 1758    C_ wss 3431
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1954  ax-ext 2431
This theorem depends on definitions:  df-bi 185  df-an 371  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-clab 2438  df-cleq 2444  df-clel 2447  df-in 3438  df-ss 3445
This theorem is referenced by:  frlmssuvc2  18340  frlmssuvc2OLD  18342
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