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Theorem elirrv 7554
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 7559 and efrirr 4555, 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 2495 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x } 
<->  x  e.  { x } ) )
2 vex 2951 . . . . 5  |-  x  e. 
_V
32snid 3833 . . . 4  |-  x  e. 
{ x }
41, 3spei 1966 . . 3  |-  E. y 
y  e.  { x }
5 snex 4397 . . . 4  |-  { x }  e.  _V
65zfregcl 7551 . . 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 3821 . . . . . . 7  |-  ( y  e.  { x }  <->  y  =  x )
9 ax-14 1729 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  x  ->  x  e.  y )
)
109equcoms 1693 . . . . . . . 8  |-  ( y  =  x  ->  (
x  e.  x  ->  x  e.  y )
)
1110com12 29 . . . . . . 7  |-  ( x  e.  x  ->  (
y  =  x  ->  x  e.  y )
)
128, 11syl5bi 209 . . . . . 6  |-  ( x  e.  x  ->  (
y  e.  { x }  ->  x  e.  y ) )
13 eleq1 2495 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  e.  { x } 
<->  x  e.  { x } ) )
1413notbid 286 . . . . . . . 8  |-  ( z  =  x  ->  ( -.  z  e.  { x } 
<->  -.  x  e.  {
x } ) )
1514rspccv 3041 . . . . . . 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 2780 . . 3  |-  ( x  e.  x  ->  A. y  e.  { x }  -.  A. z  e.  y  -.  z  e.  { x } )
20 ralnex 2707 . . 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 189 . 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 3    -> wi 4   E.wex 1550    e. wcel 1725   A.wral 2697   E.wrex 2698   {csn 3806
This theorem is referenced by:  elirr  7555  ruv  7557  dfac2  8000  nd1  8451  nd2  8452  nd3  8453  axunnd  8460  axregndlem1  8466  axregndlem2  8467  axregnd  8468  elpotr  25392  exnel  25414  distel  25415
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-14 1729  ax-6 1744  ax-7 1749  ax-11 1761  ax-12 1950  ax-ext 2416  ax-sep 4322  ax-nul 4330  ax-pr 4395  ax-reg 7549
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 2422  df-cleq 2428  df-clel 2431  df-nfc 2560  df-ne 2600  df-ral 2702  df-rex 2703  df-v 2950  df-dif 3315  df-un 3317  df-nul 3621  df-sn 3812  df-pr 3813
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