Theorem pssnel 2327

Description: A proper subclass has a member in one argument that's not in both.

Assertion

Ref	Expression
pssnel	|- (A (. B -> E.x(x e. B /\ -. x e. A))

Distinct variable groups:   x,A   x,B

Proof of Theorem pssnel

Step	Hyp	Ref	Expression
1		df-pss 2051	. . . 4 |- (A (. B <-> (A (_ B /\ A =/= B))
2		pssdifn0 2325	. . . 4 |- ((A (_ B /\ A =/= B) -> (B \ A) =/= (/))
3	1, 2	sylbi 199	. . 3 |- (A (. B -> (B \ A) =/= (/))
4		ne0 2284	. . 3 |- ((B \ A) =/= (/) <-> E.x x e. (B \ A))
5	3, 4	sylib 198	. 2 |- (A (. B -> E.x x e. (B \ A))
6		eldif 2053	. . 3 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
7	6	exbii 1049	. 2 |- (E.x x e. (B \ A) <-> E.x(x e. B /\ -. x e. A))
8	5, 7	sylib 198	1 |- (A (. B -> E.x(x e. B /\ -. x e. A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 223   e. wcel 956  E.wex 978   =/= wne 1582   \ cdif 2040   (_ wss 2043   (. wpss 2044  (/)c0 2276

This theorem is referenced by:  php 4499  php3 4501  pssnn 4519  inf3lem2 4594  genpnnp 5088  ltexprlem1 5122  reclem1pr 5136  spansncv 9537

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 960  ax-gen 961  ax-8 962  ax-10 964  ax-12 966  ax-17 969  ax-4 971  ax-5o 973  ax-6o 976  ax-9o 1121  ax-10o 1138  ax-16 1208  ax-11o 1216  ax-ext 1457

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 979  df-sb 1170  df-clab 1462  df-cleq 1467  df-clel 1470  df-ne 1584  df-v 1808  df-dif 2045  df-in 2047  df-ss 2049  df-pss 2051  df-nul 2277

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