Theorem nssne1 3215

Description: Two classes are different if they don't include the same class. (Contributed by NM, 23-Apr-2015.)

Assertion

Ref	Expression
nssne1	|- ( ( A C_ B /\ -. A C_ C ) -> B =/= C )

Proof of Theorem nssne1

Step	Hyp	Ref	Expression
1		sseq2 3181	. . . 4 |- ( B = C -> ( A C_ B <-> A C_ C ) )
2	1	biimpcd 159	. . 3 |- ( A C_ B -> ( B = C -> A C_ C ) )
3	2	necon3bd 2390	. 2 |- ( A C_ B -> ( -. A C_ C -> B =/= C ) )
4	3	imp 124	1 |- ( ( A C_ B /\ -. A C_ C ) -> B =/= C )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   = wceq 1353   =/= wne 2347   C_ wss 3131

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 614  ax-in2 615  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-11 1506  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-ne 2348  df-in 3137  df-ss 3144

This theorem is referenced by: (None)