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Theorem notzfaus 4597
Description: In the Separation Scheme zfauscl 4546, 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
StepHypRef Expression
1 notzfaus.1 . . . . . 6  |-  A  =  { (/) }
2 0ex 4554 . . . . . . 7  |-  (/)  e.  _V
32snnz 4116 . . . . . 6  |-  { (/) }  =/=  (/)
41, 3eqnetri 2721 . . . . 5  |-  A  =/=  (/)
5 n0 3772 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
64, 5mpbi 212 . . . 4  |-  E. x  x  e.  A
7 biimt 337 . . . . . 6  |-  ( x  e.  A  ->  (
x  e.  y  <->  ( x  e.  A  ->  x  e.  y ) ) )
8 iman 426 . . . . . . 7  |-  ( ( x  e.  A  ->  x  e.  y )  <->  -.  ( x  e.  A  /\  -.  x  e.  y ) )
9 notzfaus.2 . . . . . . . 8  |-  ( ph  <->  -.  x  e.  y )
109anbi2i 699 . . . . . . 7  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  -.  x  e.  y ) )
118, 10xchbinxr 313 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  y )  <->  -.  ( x  e.  A  /\  ph ) )
127, 11syl6bb 265 . . . . 5  |-  ( x  e.  A  ->  (
x  e.  y  <->  -.  (
x  e.  A  /\  ph ) ) )
13 xor3 359 . . . . 5  |-  ( -.  ( x  e.  y  <-> 
( x  e.  A  /\  ph ) )  <->  ( x  e.  y  <->  -.  ( x  e.  A  /\  ph )
) )
1412, 13sylibr 216 . . . 4  |-  ( x  e.  A  ->  -.  ( x  e.  y  <->  ( x  e.  A  /\  ph ) ) )
156, 14eximii 1705 . . 3  |-  E. x  -.  ( x  e.  y  <-> 
( x  e.  A  /\  ph ) )
16 exnal 1696 . . 3  |-  ( E. x  -.  ( x  e.  y  <->  ( x  e.  A  /\  ph )
)  <->  -.  A. x
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) )
1715, 16mpbi 212 . 2  |-  -.  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
1817nex 1675 1  |-  -.  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 188    /\ wa 371   A.wal 1436    = wceq 1438   E.wex 1660    e. wcel 1869    =/= wne 2619   (/)c0 3762   {csn 3997
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1666  ax-4 1679  ax-5 1749  ax-6 1795  ax-7 1840  ax-10 1888  ax-11 1893  ax-12 1906  ax-13 2054  ax-ext 2401  ax-nul 4553
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-tru 1441  df-ex 1661  df-nf 1665  df-sb 1788  df-clab 2409  df-cleq 2415  df-clel 2418  df-nfc 2573  df-ne 2621  df-v 3084  df-dif 3440  df-nul 3763  df-sn 3998
This theorem is referenced by: (None)
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