MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  en1uniel Structured version   Unicode version

Theorem en1uniel 7580
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 7514 . . . 4  |-  Rel  ~~
21brrelexi 5029 . . 3  |-  ( S 
~~  1o  ->  S  e. 
_V )
3 uniexg 6570 . . 3  |-  ( S  e.  _V  ->  U. S  e.  _V )
4 snidg 4042 . . 3  |-  ( U. S  e.  _V  ->  U. S  e.  { U. S } )
52, 3, 43syl 20 . 2  |-  ( S 
~~  1o  ->  U. S  e.  { U. S }
)
6 en1b 7576 . . 3  |-  ( S 
~~  1o  <->  S  =  { U. S } )
76biimpi 194 . 2  |-  ( S 
~~  1o  ->  S  =  { U. S }
)
85, 7eleqtrrd 2545 1  |-  ( S 
~~  1o  ->  U. S  e.  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1398    e. wcel 1823   _Vcvv 3106   {csn 4016   U.cuni 4235   class class class wbr 4439   1oc1o 7115    ~~ cen 7506
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1623  ax-4 1636  ax-5 1709  ax-6 1752  ax-7 1795  ax-8 1825  ax-9 1827  ax-10 1842  ax-11 1847  ax-12 1859  ax-13 2004  ax-ext 2432  ax-sep 4560  ax-nul 4568  ax-pr 4676  ax-un 6565
This theorem depends on definitions:  df-bi 185  df-or 368  df-an 369  df-3an 973  df-tru 1401  df-ex 1618  df-nf 1622  df-sb 1745  df-eu 2288  df-mo 2289  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2651  df-ral 2809  df-rex 2810  df-reu 2811  df-rab 2813  df-v 3108  df-sbc 3325  df-dif 3464  df-un 3466  df-in 3468  df-ss 3475  df-nul 3784  df-if 3930  df-sn 4017  df-pr 4019  df-op 4023  df-uni 4236  df-br 4440  df-opab 4498  df-id 4784  df-suc 4873  df-xp 4994  df-rel 4995  df-cnv 4996  df-co 4997  df-dm 4998  df-rn 4999  df-res 5000  df-ima 5001  df-iota 5534  df-fun 5572  df-fn 5573  df-f 5574  df-f1 5575  df-fo 5576  df-f1o 5577  df-fv 5578  df-1o 7122  df-en 7510
This theorem is referenced by:  en2eleq  8377  en2other2  8378  pmtrf  16682  pmtrmvd  16683  pmtrfinv  16688  frgpcyg  18788
  Copyright terms: Public domain W3C validator