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Theorem elirrv 7307
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 7312 and efrirr 4374, but this proof is direct from the Axiom of Regularity.) (Contributed by NM, 19-Aug-1993.)
Assertion
Ref Expression
elirrv  |-  -.  x  e.  x
Dummy variables  y 
z are mutually distinct and distinct from all other variables.

Proof of Theorem elirrv
StepHypRef Expression
1 eleq1 2345 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x } 
<->  x  e.  { x } ) )
2 vex 2793 . . . . 5  |-  x  e. 
_V
32snid 3669 . . . 4  |-  x  e. 
{ x }
41, 3speiv 1946 . . 3  |-  E. y 
y  e.  { x }
5 snex 4216 . . . 4  |-  { x }  e.  _V
65zfregcl 7304 . . 3  |-  ( E. y  y  e.  {
x }  ->  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x } )
74, 6ax-mp 10 . 2  |-  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x }
8 elsn 3657 . . . . . . 7  |-  ( y  e.  { x }  <->  y  =  x )
9 ax-14 1689 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  x  ->  x  e.  y )
)
109equcoms 1652 . . . . . . . 8  |-  ( y  =  x  ->  (
x  e.  x  ->  x  e.  y )
)
1110com12 29 . . . . . . 7  |-  ( x  e.  x  ->  (
y  =  x  ->  x  e.  y )
)
128, 11syl5bi 210 . . . . . 6  |-  ( x  e.  x  ->  (
y  e.  { x }  ->  x  e.  y ) )
13 eleq1 2345 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  e.  { x } 
<->  x  e.  { x } ) )
1413notbid 287 . . . . . . . 8  |-  ( z  =  x  ->  ( -.  z  e.  { x } 
<->  -.  x  e.  {
x } ) )
1514rspccv 2883 . . . . . . 7  |-  ( A. z  e.  y  -.  z  e.  { x }  ->  ( x  e.  y  ->  -.  x  e.  { x } ) )
163, 15mt2i 112 . . . . . 6  |-  ( A. z  e.  y  -.  z  e.  { x }  ->  -.  x  e.  y )
1712, 16nsyli 135 . . . . 5  |-  ( x  e.  x  ->  ( A. z  e.  y  -.  z  e.  { x }  ->  -.  y  e.  { x } ) )
1817con2d 109 . . . 4  |-  ( x  e.  x  ->  (
y  e.  { x }  ->  -.  A. z  e.  y  -.  z  e.  { x } ) )
1918ralrimiv 2627 . . 3  |-  ( x  e.  x  ->  A. y  e.  { x }  -.  A. z  e.  y  -.  z  e.  { x } )
20 ralnex 2555 . . 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 190 . 2  |-  ( x  e.  x  ->  -.  E. y  e.  { x } A. z  e.  y  -.  z  e.  {
x } )
227, 21mt2 172 1  |-  -.  x  e.  x
Colors of variables: wff set class
Syntax hints:   -. wn 5    -> wi 6   E.wex 1529    = wceq 1624    e. wcel 1685   A.wral 2545   E.wrex 2546   {csn 3642
This theorem is referenced by:  elirr  7308  ruv  7310  dfac2  7753  nd1  8205  nd2  8206  nd3  8207  axunnd  8214  axregndlem1  8220  axregndlem2  8221  axregnd  8222  elpotr  23539  distel  23562
This theorem was proved from axioms:  ax-1 7  ax-2 8  ax-3 9  ax-mp 10  ax-gen 1534  ax-5 1545  ax-17 1604  ax-9 1637  ax-8 1645  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2266  ax-sep 4143  ax-nul 4151  ax-pr 4214  ax-reg 7302
This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-tru 1312  df-ex 1530  df-nf 1533  df-sb 1632  df-clab 2272  df-cleq 2278  df-clel 2281  df-nfc 2410  df-ne 2450  df-ral 2550  df-rex 2551  df-v 2792  df-dif 3157  df-un 3159  df-nul 3458  df-sn 3648  df-pr 3649
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