Theorem pssnel 2331

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 2055	. . . 4 |- (A (. B <-> (A (_ B /\ A =/= B))
2		pssdifn0 2329	. . . 4 |- ((A (_ B /\ A =/= B) -> (B \ A) =/= (/))
3	1, 2	sylbi 199	. . 3 |- (A (. B -> (B \ A) =/= (/))
4		ne0 2288	. . 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 2057	. . 3 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
7	6	exbii 1051	. 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 958  E.wex 980   =/= wne 1585   \ cdif 2044   (_ wss 2047   (. wpss 2048  (/)c0 2280

This theorem is referenced by:  php 4513  php3 4515  php3OLD 4516  pssnn 4534  inf3lem2 4614  genpnnp 5108  ltexprlem1 5142  reclem1pr 5156  spansncv 9597

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-pss 2055  df-nul 2281

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