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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
StepHypRef Expression
1 notzfaus.1 . . . . . 6  |-  A  =  { (/) }
2 0ex 4528 . . . . . . 7  |-  (/)  e.  _V
32snnz 4092 . . . . . 6  |-  { (/) }  =/=  (/)
41, 3eqnetri 2701 . . . . 5  |-  A  =/=  (/)
5 n0 3750 . . . . 5  |-  ( A  =/=  (/)  <->  E. x  x  e.  A )
64, 5mpbi 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 )
109anbi2i 694 . . . . . . 7  |-  ( ( x  e.  A  /\  ph )  <->  ( x  e.  A  /\  -.  x  e.  y ) )
118, 10xchbinxr 311 . . . . . 6  |-  ( ( x  e.  A  ->  x  e.  y )  <->  -.  ( x  e.  A  /\  ph ) )
127, 11syl6bb 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 )
) )
1412, 13sylibr 214 . . . 4  |-  ( x  e.  A  ->  -.  ( x  e.  y  <->  ( x  e.  A  /\  ph ) ) )
156, 14eximii 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 ) ) )
1715, 16mpbi 210 . 2  |-  -.  A. x ( x  e.  y  <->  ( x  e.  A  /\  ph )
)
1817nex 1650 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 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)
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