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Theorem elirrv 7327
Description: The membership relation is irreflexive: no set is a member of itself. Theorem 105 of [Suppes] p. 54. (This is trivial to prove from zfregfr 7332 and efrirr 4390, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
elirrv  |-  -.  x  e.  x

Proof of Theorem elirrv
Dummy variables  y 
z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eleq1 2356 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x } 
<->  x  e.  { x } ) )
2 vex 2804 . . . . 5  |-  x  e. 
_V
32snid 3680 . . . 4  |-  x  e. 
{ x }
41, 3speiv 1953 . . 3  |-  E. y 
y  e.  { x }
5 snex 4232 . . . 4  |-  { x }  e.  _V
65zfregcl 7324 . . 3  |-  ( E. y  y  e.  {
x }  ->  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x } )
74, 6ax-mp 8 . 2  |-  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x }
8 elsn 3668 . . . . . . 7  |-  ( y  e.  { x }  <->  y  =  x )
9 ax-14 1700 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  x  ->  x  e.  y )
)
109equcoms 1666 . . . . . . . 8  |-  ( y  =  x  ->  (
x  e.  x  ->  x  e.  y )
)
1110com12 27 . . . . . . 7  |-  ( x  e.  x  ->  (
y  =  x  ->  x  e.  y )
)
128, 11syl5bi 208 . . . . . 6  |-  ( x  e.  x  ->  (
y  e.  { x }  ->  x  e.  y ) )
13 eleq1 2356 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  e.  { x } 
<->  x  e.  { x } ) )
1413notbid 285 . . . . . . . 8  |-  ( z  =  x  ->  ( -.  z  e.  { x } 
<->  -.  x  e.  {
x } ) )
1514rspccv 2894 . . . . . . 7  |-  ( A. z  e.  y  -.  z  e.  { x }  ->  ( x  e.  y  ->  -.  x  e.  { x } ) )
163, 15mt2i 110 . . . . . 6  |-  ( A. z  e.  y  -.  z  e.  { x }  ->  -.  x  e.  y )
1712, 16nsyli 133 . . . . 5  |-  ( x  e.  x  ->  ( A. z  e.  y  -.  z  e.  { x }  ->  -.  y  e.  { x } ) )
1817con2d 107 . . . 4  |-  ( x  e.  x  ->  (
y  e.  { x }  ->  -.  A. z  e.  y  -.  z  e.  { x } ) )
1918ralrimiv 2638 . . 3  |-  ( x  e.  x  ->  A. y  e.  { x }  -.  A. z  e.  y  -.  z  e.  { x } )
20 ralnex 2566 . . 3  |-  ( A. y  e.  { x }  -.  A. z  e.  y  -.  z  e. 
{ x }  <->  -.  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x } )
2119, 20sylib 188 . 2  |-  ( x  e.  x  ->  -.  E. y  e.  { x } A. z  e.  y  -.  z  e.  {
x } )
227, 21mt2 170 1  |-  -.  x  e.  x
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1531    = wceq 1632    e. wcel 1696   A.wral 2556   E.wrex 2557   {csn 3653
This theorem is referenced by:  elirr  7328  ruv  7330  dfac2  7773  nd1  8225  nd2  8226  nd3  8227  axunnd  8234  axregndlem1  8240  axregndlem2  8241  axregnd  8242  elpotr  24208  distel  24231
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-14 1700  ax-6 1715  ax-7 1720  ax-11 1727  ax-12 1878  ax-ext 2277  ax-sep 4157  ax-nul 4165  ax-pr 4230  ax-reg 7322
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-ral 2561  df-rex 2562  df-v 2803  df-dif 3168  df-un 3170  df-nul 3469  df-sn 3659  df-pr 3660
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