Theorem notzfaus 3478

Description: In the Separation Scheme zfauscl 3440, 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 3446	. . . . . . 7 |- (/) e. _V
2	1	snnz 3119	. . . . . 6 |- {(/)} =/= (/)
3		notzfaus.1	. . . . . . 7 |- A = {(/)}
4	3	neeq1i 2026	. . . . . 6 |- (A =/= (/) <-> {(/)} =/= (/))
5	2, 4	mpbir 207	. . . . 5 |- A =/= (/)
6		n0 2884	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
7	5, 6	mpbi 206	. . . 4 |- E.x x e. A
8		biimt 803	. . . . . . 7 |- (x e. A -> (x e. y <-> (x e. A -> x e. y)))
9		iman 256	. . . . . . . 8 |- ((x e. A -> x e. y) <-> -. (x e. A /\ -. x e. y))
10		notzfaus.2	. . . . . . . . . 10 |- (ph <-> -. x e. y)
11	10	anbi2i 538	. . . . . . . . 9 |- ((x e. A /\ ph) <-> (x e. A /\ -. x e. y))
12	11	notbii 204	. . . . . . . 8 |- (-. (x e. A /\ ph) <-> -. (x e. A /\ -. x e. y))
13	9, 12	bitr4i 193	. . . . . . 7 |- ((x e. A -> x e. y) <-> -. (x e. A /\ ph))
14	8, 13	syl6bb 595	. . . . . 6 |- (x e. A -> (x e. y <-> -. (x e. A /\ ph)))
15		xor3 737	. . . . . 6 |- (-. (x e. y <-> (x e. A /\ ph)) <-> (x e. y <-> -. (x e. A /\ ph)))
16	14, 15	sylibr 217	. . . . 5 |- (x e. A -> -. (x e. y <-> (x e. A /\ ph)))
17	16	eximi 1387	. . . 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 1385	. . 3 |- (E.x -. (x e. y <-> (x e. A /\ ph)) <-> -. A.x(x e. y <-> (x e. A /\ ph)))
20	18, 19	mpbi 206	. 2 |- -. A.x(x e. y <-> (x e. A /\ ph))
21	20	nex 1456	1 |- -. E.yA.x(x e. y <-> (x e. A /\ ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 163   /\ wa 240  A.wal 1296   = wceq 1298   e. wcel 1300  E.wex 1326   =/= wne 2017  (/)c0 2875  {csn 3044

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  ax-nul 3445

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-nul 2876  df-sn 3049

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