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Theorem notzfaus 4201
 Description: In the Separation Scheme zfauscl 4159, we require that not occur in (which can be generalized to "not be free in"). Here we show special cases of and that result in a contradiction by violating this requirement. (Contributed by NM, 8-Feb-2006.)
Hypotheses
Ref Expression
notzfaus.1
notzfaus.2
Assertion
Ref Expression
notzfaus
Distinct variable group:   ,
Allowed substitution hints:   (,)   ()

Proof of Theorem notzfaus
StepHypRef Expression
1 notzfaus.1 . . . . . 6
2 0ex 4166 . . . . . . 7
32snnz 3757 . . . . . 6
41, 3eqnetri 2476 . . . . 5
5 n0 3477 . . . . 5
64, 5mpbi 199 . . . 4
7 biimt 325 . . . . . . 7
8 iman 413 . . . . . . . 8
9 notzfaus.2 . . . . . . . . 9
109anbi2i 675 . . . . . . . 8
118, 10xchbinxr 302 . . . . . . 7
127, 11syl6bb 252 . . . . . 6
13 xor3 346 . . . . . 6
1412, 13sylibr 203 . . . . 5
1514eximi 1566 . . . 4
166, 15ax-mp 8 . . 3
17 exnal 1564 . . 3
1816, 17mpbi 199 . 2
1918nex 1545 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 176   wa 358  wal 1530  wex 1531   wceq 1632   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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