Theorem nelss 26622

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 3302	. . 3 |- ( B C_ C -> ( A e. B -> A e. C ) )
2	1	com12 29	. 2 |- ( A e. B -> ( B C_ C -> A e. C ) )
3	2	con3and 429	1 |- ( ( A e. B /\ -. A e. C ) -> -. B C_ C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 359   e. wcel 1721   C_ wss 3280

This theorem is referenced by:  frlmssuvc2  27115

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385

This theorem depends on definitions:  df-bi 178  df-an 361  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-clab 2391  df-cleq 2397  df-clel 2400  df-in 3287  df-ss 3294