Theorem pssdifn0 1750

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 1516	. . . . . 6 |- (A = B <-> (A (_ B /\ B (_ A))
2	1	biimpr 134	. . . . 5 |- ((A (_ B /\ B (_ A) -> A = B)
3	2	exp 291	. . . 4 |- (A (_ B -> (B (_ A -> A = B))
4		ssdif0 1748	. . . 4 |- (B (_ A <-> (B \ A) = (/))
5	3, 4	syl5ibr 182	. . 3 |- (A (_ B -> ((B \ A) = (/) -> A = B))
6	5	con3d 87	. 2 |- (A (_ B -> (-. A = B -> -. (B \ A) = (/)))
7	6	imp 277	1 |- ((A (_ B /\ -. A = B) -> -. (B \ A) = (/))

Syntax hints:  -. wn 1   -> wi 2   /\ wa 196   = wceq 1091   \ cdif 1484   (_ wss 1487  (/)c0 1707

This theorem is referenced by:  pssnel 1752  tz7.7 2224  inf3lem3 3466

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708

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