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Theorem ensn1g 6926
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 3651 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
21breq1d 4033 . 2  |-  ( x  =  A  ->  ( { x }  ~~  1o 
<->  { A }  ~~  1o ) )
3 vex 2791 . . 3  |-  x  e. 
_V
43ensn1 6925 . 2  |-  { x }  ~~  1o
52, 4vtoclg 2843 1  |-  ( A  e.  V  ->  { A }  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1623    e. wcel 1684   {csn 3640   class class class wbr 4023   1oc1o 6472    ~~ cen 6860
This theorem is referenced by:  enpr1g  6927  en1b  6929  en2sn  6940  snfi  6941  sucxpdom  7072  en1eqsn  7088  pr2nelem  7634  prdom2  7636  cda1en  7801  rngosn4  21094  rngoueqz  21097  snct  23339
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 1635  ax-8 1643  ax-13 1686  ax-14 1688  ax-6 1703  ax-7 1708  ax-11 1715  ax-12 1866  ax-ext 2264  ax-sep 4141  ax-nul 4149  ax-pr 4214  ax-un 4512
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 1630  df-eu 2147  df-mo 2148  df-clab 2270  df-cleq 2276  df-clel 2279  df-nfc 2408  df-ne 2448  df-ral 2548  df-rex 2549  df-rab 2552  df-v 2790  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3456  df-if 3566  df-sn 3646  df-pr 3647  df-op 3649  df-uni 3828  df-br 4024  df-opab 4078  df-id 4309  df-suc 4398  df-xp 4695  df-rel 4696  df-cnv 4697  df-co 4698  df-dm 4699  df-rn 4700  df-fun 5257  df-fn 5258  df-f 5259  df-f1 5260  df-fo 5261  df-f1o 5262  df-1o 6479  df-en 6864
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