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Theorem exnel 30009
Description: There is always a set not in  y. (Contributed by Scott Fenton, 13-Dec-2010.)
Assertion
Ref Expression
exnel  |-  E. x  -.  x  e.  y

Proof of Theorem exnel
StepHypRef Expression
1 elirrv 8056 . 2  |-  -.  y  e.  y
21nfth 1646 . . 3  |-  F/ x  -.  y  e.  y
3 ax-8 1844 . . . 4  |-  ( x  =  y  ->  (
x  e.  y  -> 
y  e.  y ) )
43con3d 133 . . 3  |-  ( x  =  y  ->  ( -.  y  e.  y  ->  -.  x  e.  y ) )
52, 4spime 2035 . 2  |-  ( -.  y  e.  y  ->  E. x  -.  x  e.  y )
61, 5ax-mp 5 1  |-  E. x  -.  x  e.  y
Colors of variables: wff setvar class
Syntax hints:   -. wn 3   E.wex 1633
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1639  ax-4 1652  ax-5 1725  ax-6 1771  ax-7 1814  ax-8 1844  ax-9 1846  ax-10 1861  ax-11 1866  ax-12 1878  ax-13 2026  ax-ext 2380  ax-sep 4516  ax-nul 4524  ax-pr 4629  ax-reg 8051
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-tru 1408  df-ex 1634  df-nf 1638  df-sb 1764  df-clab 2388  df-cleq 2394  df-clel 2397  df-nfc 2552  df-ne 2600  df-ral 2758  df-rex 2759  df-v 3060  df-dif 3416  df-un 3418  df-nul 3738  df-sn 3972  df-pr 3974
This theorem is referenced by: (None)
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