Theorem pssdifn0 3612

Description: A proper subclass has a nonempty difference. (Contributed by NM, 3-May-1994.)

Assertion

Ref	Expression
pssdifn0	|- ((A C_ B /\ A =/= B) -> (B \ A) =/= (/))

Proof of Theorem pssdifn0

Step	Hyp	Ref	Expression
1		ssdif0 3610	. . . 4 |- (B C_ A <-> (B \ A) = (/))
2		eqss 3288	. . . . 5 |- (A = B <-> (A C_ B /\ B C_ A))
3	2	simplbi2 608	. . . 4 |- (A C_ B -> (B C_ A -> A = B))
4	1, 3	syl5bir 209	. . 3 |- (A C_ B -> ((B \ A) = (/) -> A = B))
5	4	necon3d 2555	. 2 |- (A C_ B -> (A =/= B -> (B \ A) =/= (/)))
6	5	imp 418	1 |- ((A C_ B /\ A =/= B) -> (B \ A) =/= (/))

Syntax hints:   -> wi 4   /\ wa 358   = wceq 1642   =/= wne 2517   \ cdif 3207   C_ wss 3258  (/)c0 3551

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-dif 3216  df-ss 3260  df-nul 3552

This theorem is referenced by:  pssdif  3613