Theorem pssdifn0 2333

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 2080	. . . . . 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 2331	. . . 4 |- (B (_ A <-> (B \ A) = (/))
5	3, 4	syl5ibr 207	. . 3 |- (A (_ B -> ((B \ A) = (/) -> A = B))
6	5	necon3d 1607	. 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 958   =/= wne 1588   \ cdif 2047   (_ wss 2050  (/)c0 2283

This theorem is referenced by:  pssnel 2335  tz7.7 2979  inf3lem3 4624

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 964  ax-gen 965  ax-8 966  ax-10 968  ax-12 970  ax-17 973  ax-4 975  ax-5o 977  ax-6o 980  ax-9o 1125  ax-10o 1142  ax-16 1212  ax-11o 1220  ax-ext 1462

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 983  df-sb 1174  df-clab 1467  df-cleq 1472  df-clel 1475  df-ne 1590  df-v 1815  df-dif 2052  df-in 2054  df-ss 2056  df-nul 2284

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