Users' Mathboxes Mathbox for Stefan O'Rear < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  en1uniel Unicode version

Theorem en1uniel 27050
Description: A singleton contains its sole element. (Contributed by Stefan O'Rear, 16-Aug-2015.)
Assertion
Ref Expression
en1uniel  |-  ( S 
~~  1o  ->  U. S  e.  S )

Proof of Theorem en1uniel
StepHypRef Expression
1 relen 7051 . . . 4  |-  Rel  ~~
21brrelexi 4859 . . 3  |-  ( S 
~~  1o  ->  S  e. 
_V )
3 uniexg 4647 . . 3  |-  ( S  e.  _V  ->  U. S  e.  _V )
4 snidg 3783 . . 3  |-  ( U. S  e.  _V  ->  U. S  e.  { U. S } )
52, 3, 43syl 19 . 2  |-  ( S 
~~  1o  ->  U. S  e.  { U. S }
)
6 en1b 7112 . . 3  |-  ( S 
~~  1o  <->  S  =  { U. S } )
76biimpi 187 . 2  |-  ( S 
~~  1o  ->  S  =  { U. S }
)
85, 7eleqtrrd 2465 1  |-  ( S 
~~  1o  ->  U. S  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1717   _Vcvv 2900   {csn 3758   U.cuni 3958   class class class wbr 4154   1oc1o 6654    ~~ cen 7043
This theorem is referenced by:  en2eleq  27051  en2other2  27052  pmtrf  27067  pmtrmvd  27068  pmtrfinv  27072
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1661  ax-8 1682  ax-13 1719  ax-14 1721  ax-6 1736  ax-7 1741  ax-11 1753  ax-12 1939  ax-ext 2369  ax-sep 4272  ax-nul 4280  ax-pr 4345  ax-un 4642
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2243  df-mo 2244  df-clab 2375  df-cleq 2381  df-clel 2384  df-nfc 2513  df-ne 2553  df-ral 2655  df-rex 2656  df-reu 2657  df-rab 2659  df-v 2902  df-sbc 3106  df-dif 3267  df-un 3269  df-in 3271  df-ss 3278  df-nul 3573  df-if 3684  df-sn 3764  df-pr 3765  df-op 3767  df-uni 3959  df-br 4155  df-opab 4209  df-id 4440  df-suc 4529  df-xp 4825  df-rel 4826  df-cnv 4827  df-co 4828  df-dm 4829  df-rn 4830  df-res 4831  df-ima 4832  df-iota 5359  df-fun 5397  df-fn 5398  df-f 5399  df-f1 5400  df-fo 5401  df-f1o 5402  df-fv 5403  df-1o 6661  df-en 7047
  Copyright terms: Public domain W3C validator