MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  elirrv Structured version   Visualization version   Unicode version

Theorem elirrv 8130
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 8135 and efrirr 4820, 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 2537 . . . 4  |-  ( y  =  x  ->  (
y  e.  { x } 
<->  x  e.  { x } ) )
2 ssnid 3989 . . . 4  |-  x  e. 
{ x }
31, 2spei 2118 . . 3  |-  E. y 
y  e.  { x }
4 snex 4641 . . . 4  |-  { x }  e.  _V
54zfregcl 8127 . . 3  |-  ( E. y  y  e.  {
x }  ->  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x } )
63, 5ax-mp 5 . 2  |-  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x }
7 elsn 3973 . . . . . . 7  |-  ( y  e.  { x }  <->  y  =  x )
8 ax9 1917 . . . . . . . . 9  |-  ( x  =  y  ->  (
x  e.  x  ->  x  e.  y )
)
98equcoms 1872 . . . . . . . 8  |-  ( y  =  x  ->  (
x  e.  x  ->  x  e.  y )
)
109com12 31 . . . . . . 7  |-  ( x  e.  x  ->  (
y  =  x  ->  x  e.  y )
)
117, 10syl5bi 225 . . . . . 6  |-  ( x  e.  x  ->  (
y  e.  { x }  ->  x  e.  y ) )
12 eleq1 2537 . . . . . . . . 9  |-  ( z  =  x  ->  (
z  e.  { x } 
<->  x  e.  { x } ) )
1312notbid 301 . . . . . . . 8  |-  ( z  =  x  ->  ( -.  z  e.  { x } 
<->  -.  x  e.  {
x } ) )
1413rspccv 3133 . . . . . . 7  |-  ( A. z  e.  y  -.  z  e.  { x }  ->  ( x  e.  y  ->  -.  x  e.  { x } ) )
152, 14mt2i 122 . . . . . 6  |-  ( A. z  e.  y  -.  z  e.  { x }  ->  -.  x  e.  y )
1611, 15nsyli 148 . . . . 5  |-  ( x  e.  x  ->  ( A. z  e.  y  -.  z  e.  { x }  ->  -.  y  e.  { x } ) )
1716con2d 119 . . . 4  |-  ( x  e.  x  ->  (
y  e.  { x }  ->  -.  A. z  e.  y  -.  z  e.  { x } ) )
1817ralrimiv 2808 . . 3  |-  ( x  e.  x  ->  A. y  e.  { x }  -.  A. z  e.  y  -.  z  e.  { x } )
19 ralnex 2834 . . 3  |-  ( A. y  e.  { x }  -.  A. z  e.  y  -.  z  e. 
{ x }  <->  -.  E. y  e.  { x } A. z  e.  y  -.  z  e.  { x } )
2018, 19sylib 201 . 2  |-  ( x  e.  x  ->  -.  E. y  e.  { x } A. z  e.  y  -.  z  e.  {
x } )
216, 20mt2 184 1  |-  -.  x  e.  x
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4   E.wex 1671    e. wcel 1904   A.wral 2756   E.wrex 2757   {csn 3959
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pr 4639  ax-reg 8125
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-ral 2761  df-rex 2762  df-v 3033  df-dif 3393  df-un 3395  df-nul 3723  df-sn 3960  df-pr 3962
This theorem is referenced by:  elirr  8131  ruv  8133  dfac2  8579  nd1  9030  nd2  9031  nd3  9032  axunnd  9039  axregndlem1  9045  axregndlem2  9046  axregnd  9047  elpotr  30498  exnel  30520  distel  30521
  Copyright terms: Public domain W3C validator