Theorem pssnel 2321

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 2045	. . . 4 |- (A (. B <-> (A (_ B /\ A =/= B))
2		pssdifn0 2319	. . . 4 |- ((A (_ B /\ A =/= B) -> (B \ A) =/= (/))
3	1, 2	sylbi 199	. . 3 |- (A (. B -> (B \ A) =/= (/))
4		ne0 2278	. . 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 2047	. . 3 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
7	6	exbii 1047	. 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 955  E.wex 977   =/= wne 1577   \ cdif 2034   (_ wss 2037   (. wpss 2038  (/)c0 2270

This theorem is referenced by:  php 4493  php3 4495  pssnn 4513  inf3lem2 4586  genpnnp 5080  ltexprlem1 5114  reclem1pr 5128  spansncv 9514

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 959  ax-gen 960  ax-8 961  ax-10 963  ax-12 965  ax-17 968  ax-4 970  ax-5o 972  ax-6o 975  ax-9o 1119  ax-10o 1136  ax-16 1206  ax-11o 1213  ax-ext 1452

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-in 2041  df-ss 2043  df-pss 2045  df-nul 2271

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