Theorem pssnel 2938

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

Assertion

Ref	Expression
pssnel	|- (A C. 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 2607	. . . 4 |- (A C. B <-> (A C_ B /\ A =/= B))
2		pssdifn0 2936	. . . 4 |- ((A C_ B /\ A =/= B) -> (B \ A) =/= (/))
3	1, 2	sylbi 216	. . 3 |- (A C. B -> (B \ A) =/= (/))
4		n0 2884	. . 3 |- ((B \ A) =/= (/) <-> E.x x e. (B \ A))
5	3, 4	sylib 215	. 2 |- (A C. B -> E.x x e. (B \ A))
6		eldif 2609	. . 3 |- (x e. (B \ A) <-> (x e. B /\ -. x e. A))
7	6	exbii 1398	. 2 |- (E.x x e. (B \ A) <-> E.x(x e. B /\ -. x e. A))
8	5, 7	sylib 215	1 |- (A C. B -> E.x(x e. B /\ -. x e. A))

Syntax hints:  -. wn 2   -> wi 3   /\ wa 240   e. wcel 1300  E.wex 1326   =/= wne 2017   \ cdif 2590   C_ wss 2593   C. wpss 2594  (/)c0 2875

This theorem is referenced by:  php 5607  php3 5609  pssnn 5628  inf3lem2 5720  genpnnp 6260  ltexprlem1 6294  reclem1pr 6308  spansncvi 11232  osumcllem11 17374  pexmidlem8 17385

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 1304  ax-gen 1305  ax-8 1306  ax-9 1307  ax-10 1308  ax-11 1309  ax-12 1310  ax-17 1317  ax-4 1319  ax-5o 1321  ax-6o 1324  ax-9o 1481  ax-10o 1500  ax-16 1580  ax-11o 1588  ax-ext 1865

This theorem depends on definitions:  df-bi 164  df-or 241  df-an 242  df-ex 1327  df-sb 1536  df-clab 1872  df-cleq 1877  df-clel 1880  df-ne 2019  df-v 2294  df-dif 2597  df-in 2603  df-ss 2605  df-pss 2607  df-nul 2876

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