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Theorem notzfaus 4374
Description: In the Separation Scheme zfauscl 4332, 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 4339 . . . . . . 7  |-  (/)  e.  _V
32snnz 3922 . . . . . 6  |-  { (/) }  =/=  (/)
41, 3eqnetri 2618 . . . . 5  |-  A  =/=  (/)
5 n0 3637 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
64, 5mpbi 200 . . . 4  |-  E. x  x  e.  A
7 biimt 326 . . . . . 6  |-  ( x  e.  A  ->  (
x  e.  y  <->  ( x  e.  A  ->  x  e.  y ) ) )
8 iman 414 . . . . . . 7  |-  ( ( x  e.  A  ->  x  e.  y )  <->  -.  ( x  e.  A  /\  -.  x  e.  y ) )
9 notzfaus.2 . . . . . . . 8  |-  ( ph  <->  -.  x  e.  y )
109anbi2i 676 . . . . . . 7  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  -.  x  e.  y ) )
118, 10xchbinxr 303 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  y )  <->  -.  ( x  e.  A  /\  ph ) )
127, 11syl6bb 253 . . . . 5  |-  ( x  e.  A  ->  (
x  e.  y  <->  -.  (
x  e.  A  /\  ph ) ) )
13 xor3 347 . . . . 5  |-  ( -.  ( x  e.  y  <-> 
( x  e.  A  /\  ph ) )  <->  ( x  e.  y  <->  -.  ( x  e.  A  /\  ph )
) )
1412, 13sylibr 204 . . . 4  |-  ( x  e.  A  ->  -.  ( x  e.  y  <->  ( x  e.  A  /\  ph ) ) )
156, 14eximii 1587 . . 3  |-  E. x  -.  ( x  e.  y  <-> 
( x  e.  A  /\  ph ) )
16 exnal 1583 . . 3  |-  ( E. x  -.  ( x  e.  y  <->  ( x  e.  A  /\  ph )
)  <->  -.  A. x
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) )
1715, 16mpbi 200 . 2  |-  -.  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
1817nex 1564 1  |-  -.  E. y A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    <-> wb 177    /\ wa 359   A.wal 1549   E.wex 1550    = wceq 1652    e. wcel 1725    =/= wne 2599   (/)c0 3628   {csn 3814
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1555  ax-5 1566  ax-17 1626  ax-9 1666  ax-8 1687  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2417  ax-nul 4338
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-tru 1328  df-ex 1551  df-nf 1554  df-sb 1659  df-clab 2423  df-cleq 2429  df-clel 2432  df-nfc 2561  df-ne 2601  df-v 2958  df-dif 3323  df-nul 3629  df-sn 3820
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