Theorem notzfaus 4571

Description: In the Separation Scheme zfauscl 4521, we require that y not occur in ph (which can be generalized to "not be free in"). Here we show special cases of A and ph that result in a contradiction by violating this requirement. (Contributed by NM, 8-Feb-2006.)

Hypotheses

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

Assertion

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

Distinct variable group:   x, A

Allowed substitution hints:   ph( x, y)   A( y)

Proof of Theorem notzfaus

Step	Hyp	Ref	Expression
1		notzfaus.1	. . . . . 6 |- A = { (/) }
2		0ex 4528	. . . . . . 7 |- (/) e. _V
3	2	snnz 4092	. . . . . 6 |- { (/) } =/= (/)
4	1, 3	eqnetri 2701	. . . . 5 |- A =/= (/)
5		n0 3750	. . . . 5 |- ( A =/= (/) <-> E. x x e. A )
6	4, 5	mpbi 210	. . . 4 |- E. x x e. A
7		biimt 335	. . . . . 6 |- ( x e. A -> ( x e. y <-> ( x e. A -> x e. y ) ) )
8		iman 424	. . . . . . 7 |- ( ( x e. A -> x e. y ) <-> -. ( x e. A /\ -. x e. y ) )
9		notzfaus.2	. . . . . . . 8 |- ( ph <-> -. x e. y )
10	9	anbi2i 694	. . . . . . 7 |- ( ( x e. A /\ ph ) <-> ( x e. A /\ -. x e. y ) )
11	8, 10	xchbinxr 311	. . . . . 6 |- ( ( x e. A -> x e. y ) <-> -. ( x e. A /\ ph ) )
12	7, 11	syl6bb 263	. . . . 5 |- ( x e. A -> ( x e. y <-> -. ( x e. A /\ ph ) ) )
13		xor3 357	. . . . 5 |- ( -. ( x e. y <-> ( x e. A /\ ph ) ) <-> ( x e. y <-> -. ( x e. A /\ ph ) ) )
14	12, 13	sylibr 214	. . . 4 |- ( x e. A -> -. ( x e. y <-> ( x e. A /\ ph ) ) )
15	6, 14	eximii 1681	. . 3 |- E. x -. ( x e. y <-> ( x e. A /\ ph ) )
16		exnal 1671	. . 3 |- ( E. x -. ( x e. y <-> ( x e. A /\ ph ) ) <-> -. A. x ( x e. y <-> ( x e. A /\ ph ) ) )
17	15, 16	mpbi 210	. 2 |- -. A. x ( x e. y <-> ( x e. A /\ ph ) )
18	17	nex 1650	1 |- -. E. y A. x ( x e. y <-> ( x e. A /\ ph ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 186   /\ wa 369   A.wal 1405   = wceq 1407   E.wex 1635   e. wcel 1844   =/= wne 2600   (/)c0 3740   {csn 3974

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1641  ax-4 1654  ax-5 1727  ax-6 1773  ax-7 1816  ax-10 1863  ax-11 1868  ax-12 1880  ax-13 2028  ax-ext 2382  ax-nul 4527

This theorem depends on definitions:  df-bi 187  df-or 370  df-an 371  df-tru 1410  df-ex 1636  df-nf 1640  df-sb 1766  df-clab 2390  df-cleq 2396  df-clel 2399  df-nfc 2554  df-ne 2602  df-v 3063  df-dif 3419  df-nul 3741  df-sn 3975

This theorem is referenced by: (None)