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Theorem en1uniel 7492
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 7426 . . . 4  |-  Rel  ~~
21brrelexi 4988 . . 3  |-  ( S 
~~  1o  ->  S  e. 
_V )
3 uniexg 6488 . . 3  |-  ( S  e.  _V  ->  U. S  e.  _V )
4 snidg 4012 . . 3  |-  ( U. S  e.  _V  ->  U. S  e.  { U. S } )
52, 3, 43syl 20 . 2  |-  ( S 
~~  1o  ->  U. S  e.  { U. S }
)
6 en1b 7488 . . 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 1370    e. wcel 1758   _Vcvv 3078   {csn 3986   U.cuni 4200   class class class wbr 4401   1oc1o 7024    ~~ cen 7418
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1592  ax-4 1603  ax-5 1671  ax-6 1710  ax-7 1730  ax-8 1760  ax-9 1762  ax-10 1777  ax-11 1782  ax-12 1794  ax-13 1955  ax-ext 2432  ax-sep 4522  ax-nul 4530  ax-pr 4640  ax-un 6483
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 967  df-tru 1373  df-ex 1588  df-nf 1591  df-sb 1703  df-eu 2266  df-mo 2267  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2604  df-ne 2650  df-ral 2804  df-rex 2805  df-reu 2806  df-rab 2808  df-v 3080  df-sbc 3295  df-dif 3440  df-un 3442  df-in 3444  df-ss 3451  df-nul 3747  df-if 3901  df-sn 3987  df-pr 3989  df-op 3993  df-uni 4201  df-br 4402  df-opab 4460  df-id 4745  df-suc 4834  df-xp 4955  df-rel 4956  df-cnv 4957  df-co 4958  df-dm 4959  df-rn 4960  df-res 4961  df-ima 4962  df-iota 5490  df-fun 5529  df-fn 5530  df-f 5531  df-f1 5532  df-fo 5533  df-f1o 5534  df-fv 5535  df-1o 7031  df-en 7422
This theorem is referenced by:  en2eleq  8287  en2other2  8288  pmtrf  16081  pmtrmvd  16082  pmtrfinv  16087  frgpcyg  18132
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