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

Theorem moabex 4715
Description: "At most one" existence implies a class abstraction exists. (Contributed by NM, 30-Dec-1996.)
Assertion
Ref Expression
moabex  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )

Proof of Theorem moabex
Dummy variable  y is distinct from all other variables.
StepHypRef Expression
1 mo2v 2290 . 2  |-  ( E* x ph  <->  E. y A. x ( ph  ->  x  =  y ) )
2 abss 3565 . . . . 5  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  e. 
{ y } ) )
3 elsn 4046 . . . . . . 7  |-  ( x  e.  { y }  <-> 
x  =  y )
43imbi2i 312 . . . . . 6  |-  ( (
ph  ->  x  e.  {
y } )  <->  ( ph  ->  x  =  y ) )
54albii 1641 . . . . 5  |-  ( A. x ( ph  ->  x  e.  { y } )  <->  A. x ( ph  ->  x  =  y ) )
62, 5bitri 249 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  <->  A. x
( ph  ->  x  =  y ) )
7 snex 4697 . . . . 5  |-  { y }  e.  _V
87ssex 4600 . . . 4  |-  ( { x  |  ph }  C_ 
{ y }  ->  { x  |  ph }  e.  _V )
96, 8sylbir 213 . . 3  |-  ( A. x ( ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
109exlimiv 1723 . 2  |-  ( E. y A. x (
ph  ->  x  =  y )  ->  { x  |  ph }  e.  _V )
111, 10sylbi 195 1  |-  ( E* x ph  ->  { x  |  ph }  e.  _V )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4   A.wal 1393   E.wex 1613    e. wcel 1819   E*wmo 2284   {cab 2442   _Vcvv 3109    C_ wss 3471   {csn 4032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1619  ax-4 1632  ax-5 1705  ax-6 1748  ax-7 1791  ax-9 1823  ax-10 1838  ax-11 1843  ax-12 1855  ax-13 2000  ax-ext 2435  ax-sep 4578  ax-nul 4586  ax-pr 4695
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-tru 1398  df-ex 1614  df-nf 1618  df-sb 1741  df-eu 2287  df-mo 2288  df-clab 2443  df-cleq 2449  df-clel 2452  df-nfc 2607  df-ne 2654  df-v 3111  df-dif 3474  df-un 3476  df-in 3478  df-ss 3485  df-nul 3794  df-sn 4033  df-pr 4035
This theorem is referenced by:  rmorabex  4716  euabex  4717
  Copyright terms: Public domain W3C validator