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Theorem ensn1g 7578
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
Assertion
Ref Expression
ensn1g  |-  ( A  e.  V  ->  { A }  ~~  1o )

Proof of Theorem ensn1g
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 sneq 4020 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
21breq1d 4443 . 2  |-  ( x  =  A  ->  ( { x }  ~~  1o 
<->  { A }  ~~  1o ) )
3 vex 3096 . . 3  |-  x  e. 
_V
43ensn1 7577 . 2  |-  { x }  ~~  1o
52, 4vtoclg 3151 1  |-  ( A  e.  V  ->  { A }  ~~  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1381    e. wcel 1802   {csn 4010   class class class wbr 4433   1oc1o 7121    ~~ cen 7511
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1603  ax-4 1616  ax-5 1689  ax-6 1732  ax-7 1774  ax-8 1804  ax-9 1806  ax-10 1821  ax-11 1826  ax-12 1838  ax-13 1983  ax-ext 2419  ax-sep 4554  ax-nul 4562  ax-pr 4672  ax-un 6573
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 974  df-tru 1384  df-ex 1598  df-nf 1602  df-sb 1725  df-eu 2270  df-mo 2271  df-clab 2427  df-cleq 2433  df-clel 2436  df-nfc 2591  df-ne 2638  df-ral 2796  df-rex 2797  df-rab 2800  df-v 3095  df-dif 3461  df-un 3463  df-in 3465  df-ss 3472  df-nul 3768  df-if 3923  df-sn 4011  df-pr 4013  df-op 4017  df-uni 4231  df-br 4434  df-opab 4492  df-id 4781  df-suc 4870  df-xp 4991  df-rel 4992  df-cnv 4993  df-co 4994  df-dm 4995  df-rn 4996  df-fun 5576  df-fn 5577  df-f 5578  df-f1 5579  df-fo 5580  df-f1o 5581  df-1o 7128  df-en 7515
This theorem is referenced by:  enpr1g  7579  en1b  7581  en2sn  7593  snfi  7594  snnen2o  7704  sucxpdom  7727  en1eqsn  7747  en1eqsnbi  7748  pr2nelem  8380  prdom2  8382  cda1en  8553  rngoueqz  25297  snct  27399
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