Theorem nssne1 3328

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 3294	. . . 4 |- (B = C -> (A C_ B <-> A C_ C))
2	1	biimpcd 215	. . 3 |- (A C_ B -> (B = C -> A C_ C))
3	2	necon3bd 2554	. 2 |- (A C_ B -> (-. A C_ C -> B =/= C))
4	3	imp 418	1 |- ((A C_ B /\ -. A C_ C) -> B =/= C)

Syntax hints:  -. wn 3   -> wi 4   /\ wa 358   = wceq 1642   =/= wne 2517   C_ wss 3258

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-ne 2519  df-v 2862  df-nin 3212  df-compl 3213  df-in 3214  df-ss 3260

This theorem is referenced by: (None)