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Theorem notzfaus 4201
Description: In the Separation Scheme zfauscl 4159, 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 4166 . . . . . . 7  |-  (/)  e.  _V
32snnz 3757 . . . . . 6  |-  { (/) }  =/=  (/)
41, 3eqnetri 2476 . . . . 5  |-  A  =/=  (/)
5 n0 3477 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
64, 5mpbi 199 . . . 4  |-  E. x  x  e.  A
7 biimt 325 . . . . . . 7  |-  ( x  e.  A  ->  (
x  e.  y  <->  ( x  e.  A  ->  x  e.  y ) ) )
8 iman 413 . . . . . . . 8  |-  ( ( x  e.  A  ->  x  e.  y )  <->  -.  ( x  e.  A  /\  -.  x  e.  y ) )
9 notzfaus.2 . . . . . . . . 9  |-  ( ph  <->  -.  x  e.  y )
109anbi2i 675 . . . . . . . 8  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  -.  x  e.  y ) )
118, 10xchbinxr 302 . . . . . . 7  |-  ( ( x  e.  A  ->  x  e.  y )  <->  -.  ( x  e.  A  /\  ph ) )
127, 11syl6bb 252 . . . . . 6  |-  ( x  e.  A  ->  (
x  e.  y  <->  -.  (
x  e.  A  /\  ph ) ) )
13 xor3 346 . . . . . 6  |-  ( -.  ( x  e.  y  <-> 
( x  e.  A  /\  ph ) )  <->  ( x  e.  y  <->  -.  ( x  e.  A  /\  ph )
) )
1412, 13sylibr 203 . . . . 5  |-  ( x  e.  A  ->  -.  ( x  e.  y  <->  ( x  e.  A  /\  ph ) ) )
1514eximi 1566 . . . 4  |-  ( E. x  x  e.  A  ->  E. x  -.  (
x  e.  y  <->  ( x  e.  A  /\  ph )
) )
166, 15ax-mp 8 . . 3  |-  E. x  -.  ( x  e.  y  <-> 
( x  e.  A  /\  ph ) )
17 exnal 1564 . . 3  |-  ( E. x  -.  ( x  e.  y  <->  ( x  e.  A  /\  ph )
)  <->  -.  A. x
( x  e.  y  <-> 
( x  e.  A  /\  ph ) ) )
1816, 17mpbi 199 . 2  |-  -.  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
1918nex 1545 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 176    /\ wa 358   A.wal 1530   E.wex 1531    = wceq 1632    e. wcel 1696    =/= wne 2459   (/)c0 3468   {csn 3653
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1536  ax-5 1547  ax-17 1606  ax-9 1644  ax-8 1661  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-nul 4165
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-tru 1310  df-ex 1532  df-nf 1535  df-sb 1639  df-clab 2283  df-cleq 2289  df-clel 2292  df-nfc 2421  df-ne 2461  df-v 2803  df-dif 3168  df-nul 3469  df-sn 3659
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