Theorem pssdifn0 2329

Description: A proper subclass has a nonempty difference.

Assertion

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

Proof of Theorem pssdifn0

Step	Hyp	Ref	Expression
1		eqss 2077	. . . . . 6 |- (A = B <-> (A (_ B /\ B (_ A))
2	1	biimpr 152	. . . . 5 |- ((A (_ B /\ B (_ A) -> A = B)
3	2	ex 373	. . . 4 |- (A (_ B -> (B (_ A -> A = B))
4		ssdif0 2327	. . . 4 |- (B (_ A <-> (B \ A) = (/))
5	3, 4	syl5ibr 207	. . 3 |- (A (_ B -> ((B \ A) = (/) -> A = B))
6	5	necon3d 1604	. 2 |- (A (_ B -> (A =/= B -> (B \ A) =/= (/)))
7	6	imp 350	1 |- ((A (_ B /\ A =/= B) -> (B \ A) =/= (/))

Syntax hints:   -> wi 3   /\ wa 223   = wceq 956   =/= wne 1585   \ cdif 2044   (_ wss 2047  (/)c0 2280

This theorem is referenced by:  pssnel 2331  tz7.7 2973  inf3lem3 4615

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-in 2051  df-ss 2053  df-nul 2281

Copyright terms: Public domain