Theorem nelss 3240

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 3173	. . 3 |- ( B C_ C -> ( A e. B -> A e. C ) )
2	1	com12 30	. 2 |- ( A e. B -> ( B C_ C -> A e. C ) )
3	2	con3dimp 636	1 |- ( ( A e. B /\ -. A e. C ) -> -. B C_ C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   e. wcel 2164   C_ wss 3153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-11 1517  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-in 3159  df-ss 3166

This theorem is referenced by: (None)