Theorem notzfaus 2815

Description: In the Separation Scheme zfauscl 2779, 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 2785	. . . . . . 7 |- (/) e. V
2	1	snnz 2522	. . . . . 6 |- {(/)} =/= (/)
3		notzfaus.1	. . . . . . 7 |- A = {(/)}
4	3	neeq1i 1635	. . . . . 6 |- (A =/= (/) <-> {(/)} =/= (/))
5	2, 4	mpbir 188	. . . . 5 |- A =/= (/)
6		n0 2341	. . . . 5 |- (A =/= (/) <-> E.x x e. A)
7	5, 6	mpbi 187	. . . 4 |- E.x x e. A
8		biimt 736	. . . . . . 7 |- (x e. A -> (x e. y <-> (x e. A -> x e. y)))
9		iman 235	. . . . . . . 8 |- ((x e. A -> x e. y) <-> -. (x e. A /\ -. x e. y))
10		notzfaus.2	. . . . . . . . . 10 |- (ph <-> -. x e. y)
11	10	anbi2i 483	. . . . . . . . 9 |- ((x e. A /\ ph) <-> (x e. A /\ -. x e. y))
12	11	notbii 185	. . . . . . . 8 |- (-. (x e. A /\ ph) <-> -. (x e. A /\ -. x e. y))
13	9, 12	bitr4i 174	. . . . . . 7 |- ((x e. A -> x e. y) <-> -. (x e. A /\ ph))
14	8, 13	syl6bb 539	. . . . . 6 |- (x e. A -> (x e. y <-> -. (x e. A /\ ph)))
15		xor3 677	. . . . . 6 |- (-. (x e. y <-> (x e. A /\ ph)) <-> (x e. y <-> -. (x e. A /\ ph)))
16	14, 15	sylibr 198	. . . . 5 |- (x e. A -> -. (x e. y <-> (x e. A /\ ph)))
17	16	19.22i 1076	. . . 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 1074	. . 3 |- (E.x -. (x e. y <-> (x e. A /\ ph)) <-> -. A.x(x e. y <-> (x e. A /\ ph)))
20	18, 19	mpbi 187	. 2 |- -. A.x(x e. y <-> (x e. A /\ ph))
21	20	nex 1137	1 |- -. E.yA.x(x e. y <-> (x e. A /\ ph))

Syntax hints:  -. wn 2   -> wi 3   <-> wb 144   /\ wa 221  A.wal 990   = wceq 992   e. wcel 994  E.wex 1016   =/= wne 1628  (/)c0 2332  {csn 2467

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 998  ax-gen 999  ax-8 1000  ax-10 1002  ax-11 1003  ax-12 1004  ax-14 1006  ax-17 1007  ax-4 1009  ax-5o 1011  ax-6o 1014  ax-9o 1159  ax-10o 1177  ax-16 1247  ax-11o 1255  ax-ext 1500  ax-nul 2784

This theorem depends on definitions:  df-bi 145  df-or 222  df-an 223  df-ex 1017  df-sb 1209  df-eu 1421  df-mo 1422  df-clab 1506  df-cleq 1511  df-clel 1514  df-ne 1630  df-v 1858  df-dif 2101  df-un 2102  df-nul 2333  df-sn 2470  df-pr 2471

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