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

Step	Hyp	Ref	Expression
1		ssel 3453	. . 3 |- ( B C_ C -> ( A e. B -> A e. C ) )
2	1	com12 31	. 2 |- ( A e. B -> ( B C_ C -> A e. C ) )
3	2	con3dimp 441	1 |- ( ( A e. B /\ -. A e. C ) -> -. B C_ C )

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