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Theorem ensn1g 6594
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 3477 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
21breq1d 3877 . 2  |-  ( x  =  A  ->  ( { x }  ~~  1o 
<->  { A }  ~~  1o ) )
3 vex 2636 . . 3  |-  x  e. 
_V
43ensn1 6593 . 2  |-  { x }  ~~  1o
52, 4vtoclg 2693 1  |-  ( A  e.  V  ->  { A }  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1296    e. wcel 1445   {csn 3466   class class class wbr 3867   1oc1o 6212    ~~ cen 6535
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-13 1456  ax-14 1457  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077  ax-sep 3978  ax-nul 3986  ax-pow 4030  ax-pr 4060  ax-un 4284
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-eu 1958  df-mo 1959  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-rex 2376  df-v 2635  df-dif 3015  df-un 3017  df-in 3019  df-ss 3026  df-nul 3303  df-pw 3451  df-sn 3472  df-pr 3473  df-op 3475  df-uni 3676  df-br 3868  df-opab 3922  df-id 4144  df-suc 4222  df-xp 4473  df-rel 4474  df-cnv 4475  df-co 4476  df-dm 4477  df-rn 4478  df-fun 5051  df-fn 5052  df-f 5053  df-f1 5054  df-fo 5055  df-f1o 5056  df-1o 6219  df-en 6538
This theorem is referenced by:  enpr1g  6595  en1bg  6597  en2sn  6610  snfig  6611  snnen2og  6655  en1eqsn  6737  en1eqsnbi  6738  pr2nelem  6916
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