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Theorem nelss 26150
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 3175 . . 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 1685    C_ wss 3153
This theorem is referenced by:  frlmssuvc2  26646
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1867  ax-ext 2265
This theorem depends on definitions:  df-bi 179  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2271  df-cleq 2277  df-clel 2280  df-in 3160  df-ss 3167
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