Theorem notzfaus 2741

Description: In the Separation Scheme zfauscl 2705, 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 2711	. . . . . . 7 |- (/) e. V
2	1	snnz 2458	. . . . . 6 |- {(/)} =/= (/)
3		notzfaus.1	. . . . . . 7 |- A = {(/)}
4	3	neeq1i 1592	. . . . . 6 |- (A =/= (/) <-> {(/)} =/= (/))
5	2, 4	mpbir 190	. . . . 5 |- A =/= (/)
6		ne0 2288	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
7	5, 6	mpbi 189	. . . 4 |- E.x x e. A
8		biimt 731	. . . . . . 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 480	. . . . . . . . 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 536	. . . . . 6 |- (x e. A -> (x e. y <-> -. (x e. A /\ ph)))
15		xor3 674	. . . . . 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 1040	. . . 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 1038	. . 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 1101	1 |- -. E.yA.x(x e. y <-> (x e. A /\ ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 146   /\ wa 223  A.wal 954   = wceq 956   e. wcel 958  E.wex 980   =/= wne 1585  (/)c0 2280  {csn 2409

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-11 967  ax-12 968  ax-14 970  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  ax-nul 2710

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-v 1812  df-dif 2049  df-un 2050  df-nul 2281  df-sn 2412  df-pr 2413

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