Theorem notzfaus 2731

Description: In the Separation Scheme zfauscl 2695, we require that y not occur in ph (which can be generalized to "not be free in"). Here we show that a contradiction can result if we omit this requirement.

Hypotheses

Ref	Expression
notzfaus.1	|- A = {(/)}
notzfaus.2	|- (ph <-> -. x e. y)

Assertion

Ref	Expression
notzfaus	|- -. E.yA.x(x e. y <-> (x e. A /\ ph))

Distinct variable group:   x,A

Proof of Theorem notzfaus

Step	Hyp	Ref	Expression
1		0ex 2701	. . . . . . 7 |- (/) e. V
2	1	snnz 2449	. . . . . 6 |- {(/)} =/= (/)
3		notzfaus.1	. . . . . . 7 |- A = {(/)}
4	3	neeq1i 1584	. . . . . 6 |- (A =/= (/) <-> {(/)} =/= (/))
5	2, 4	mpbir 190	. . . . 5 |- A =/= (/)
6		ne0 2278	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
7	5, 6	mpbi 189	. . . 4 |- E.x x e. A
8		biimt 729	. . . . . . 7 |- (x e. A -> (x e. y <-> (x e. A -> x e. y)))
9		iman 237	. . . . . . . 8 |- ((x e. A -> x e. y) <-> -. (x e. A /\ -. x e. y))
10		notzfaus.2	. . . . . . . . . 10 |- (ph <-> -. x e. y)
11	10	anbi2i 479	. . . . . . . . 9 |- ((x e. A /\ ph) <-> (x e. A /\ -. x e. y))
12	11	negbii 187	. . . . . . . 8 |- (-. (x e. A /\ ph) <-> -. (x e. A /\ -. x e. y))
13	9, 12	bitr4 176	. . . . . . 7 |- ((x e. A -> x e. y) <-> -. (x e. A /\ ph))
14	8, 13	syl6bb 534	. . . . . 6 |- (x e. A -> (x e. y <-> -. (x e. A /\ ph)))
15		xor3 672	. . . . . 6 |- (-. (x e. y <-> (x e. A /\ ph)) <-> (x e. y <-> -. (x e. A /\ ph)))
16	14, 15	sylibr 200	. . . . 5 |- (x e. A -> -. (x e. y <-> (x e. A /\ ph)))
17	16	19.22i 1036	. . . 4 |- (E.x x e. A -> E.x -. (x e. y <-> (x e. A /\ ph)))
18	7, 17	ax-mp 7	. . 3 |- E.x -. (x e. y <-> (x e. A /\ ph))
19		exnal 1034	. . 3 |- (E.x -. (x e. y <-> (x e. A /\ ph)) <-> -. A.x(x e. y <-> (x e. A /\ ph)))
20	18, 19	mpbi 189	. 2 |- -. A.x(x e. y <-> (x e. A /\ ph))
21	20	nex 1097	1 |- -. E.yA.x(x e. y <-> (x e. A /\ ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 951   = wceq 953   e. wcel 955  E.wex 977   =/= wne 1577  (/)c0 2270  {csn 2399

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-11 964  ax-12 965  ax-14 967  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  ax-nul 2700

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 978  df-sb 1168  df-eu 1375  df-mo 1376  df-clab 1457  df-cleq 1462  df-clel 1465  df-ne 1579  df-v 1803  df-dif 2039  df-un 2040  df-nul 2271  df-sn 2402  df-pr 2403

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