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Theorem notzfaus 4374
 Description: In the Separation Scheme zfauscl 4332, 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 4339 . . . . . . 7
32snnz 3922 . . . . . 6
41, 3eqnetri 2618 . . . . 5
5 n0 3637 . . . . 5
64, 5mpbi 200 . . . 4
7 biimt 326 . . . . . 6
8 iman 414 . . . . . . 7
9 notzfaus.2 . . . . . . . 8
109anbi2i 676 . . . . . . 7
118, 10xchbinxr 303 . . . . . 6
127, 11syl6bb 253 . . . . 5
13 xor3 347 . . . . 5
1412, 13sylibr 204 . . . 4
156, 14eximii 1587 . . 3
16 exnal 1583 . . 3
1715, 16mpbi 200 . 2
1817nex 1564 1
 Colors of variables: wff set class Syntax hints:   wn 3   wi 4   wb 177   wa 359  wal 1549  wex 1550   wceq 1652   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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