ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  rext Unicode version

Theorem rext 3979
Description: A theorem similar to extensionality, requiring the existence of a singleton. Exercise 8 of [TakeutiZaring] p. 16. (Contributed by NM, 10-Aug-1993.)
Assertion
Ref Expression
rext  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
Distinct variable group:    x, y, z

Proof of Theorem rext
StepHypRef Expression
1 vex 2577 . . . 4  |-  x  e. 
_V
21snid 3430 . . 3  |-  x  e. 
{ x }
3 snexgOLD 3963 . . . . 5  |-  ( x  e.  _V  ->  { x }  e.  _V )
41, 3ax-mp 7 . . . 4  |-  { x }  e.  _V
5 eleq2 2117 . . . . 5  |-  ( z  =  { x }  ->  ( x  e.  z  <-> 
x  e.  { x } ) )
6 eleq2 2117 . . . . 5  |-  ( z  =  { x }  ->  ( y  e.  z  <-> 
y  e.  { x } ) )
75, 6imbi12d 227 . . . 4  |-  ( z  =  { x }  ->  ( ( x  e.  z  ->  y  e.  z )  <->  ( x  e.  { x }  ->  y  e.  { x }
) ) )
84, 7spcv 2663 . . 3  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  (
x  e.  { x }  ->  y  e.  {
x } ) )
92, 8mpi 15 . 2  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  y  e.  { x } )
10 velsn 3420 . . 3  |-  ( y  e.  { x }  <->  y  =  x )
11 equcomi 1608 . . 3  |-  ( y  =  x  ->  x  =  y )
1210, 11sylbi 118 . 2  |-  ( y  e.  { x }  ->  x  =  y )
139, 12syl 14 1  |-  ( A. z ( x  e.  z  ->  y  e.  z )  ->  x  =  y )
Colors of variables: wff set class
Syntax hints:    -> wi 4   A.wal 1257    = wceq 1259    e. wcel 1409   _Vcvv 2574   {csn 3403
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3903  ax-pow 3955
This theorem depends on definitions:  df-bi 114  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-v 2576  df-in 2952  df-ss 2959  df-pw 3389  df-sn 3409
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator