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Theorem en1uniel 26791
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 6864 . . . 4  |-  Rel  ~~
21brrelexi 4728 . . 3  |-  ( S 
~~  1o  ->  S  e. 
_V )
3 uniexg 4516 . . 3  |-  ( S  e.  _V  ->  U. S  e.  _V )
4 snidg 3666 . . 3  |-  ( U. S  e.  _V  ->  U. S  e.  { U. S } )
52, 3, 43syl 18 . 2  |-  ( S 
~~  1o  ->  U. S  e.  { U. S }
)
6 en1b 6925 . . 3  |-  ( S 
~~  1o  <->  S  =  { U. S } )
76biimpi 186 . 2  |-  ( S 
~~  1o  ->  S  =  { U. S }
)
85, 7eleqtrrd 2361 1  |-  ( S 
~~  1o  ->  U. S  e.  S )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1685   _Vcvv 2789   {csn 3641   U.cuni 3828   class class class wbr 4024   1oc1o 6468    ~~ cen 6856
This theorem is referenced by:  en2eleq  26792  en2other2  26793  pmtrf  26808  pmtrmvd  26809  pmtrfinv  26813
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1533  ax-5 1544  ax-17 1603  ax-9 1636  ax-8 1644  ax-13 1687  ax-14 1689  ax-6 1704  ax-7 1709  ax-11 1716  ax-12 1868  ax-ext 2265  ax-sep 4142  ax-nul 4150  ax-pr 4213  ax-un 4511
This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-3an 936  df-tru 1310  df-ex 1529  df-nf 1532  df-sb 1631  df-eu 2148  df-mo 2149  df-clab 2271  df-cleq 2277  df-clel 2280  df-nfc 2409  df-ne 2449  df-ral 2549  df-rex 2550  df-reu 2551  df-rab 2553  df-v 2791  df-sbc 2993  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-nul 3457  df-if 3567  df-sn 3647  df-pr 3648  df-op 3650  df-uni 3829  df-br 4025  df-opab 4079  df-id 4308  df-suc 4397  df-xp 4694  df-rel 4695  df-cnv 4696  df-co 4697  df-dm 4698  df-rn 4699  df-res 4700  df-ima 4701  df-fun 5223  df-fn 5224  df-f 5225  df-f1 5226  df-fo 5227  df-f1o 5228  df-fv 5229  df-1o 6475  df-en 6860
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