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Theorem ensn1g 7581
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 4037 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
21breq1d 4457 . 2  |-  ( x  =  A  ->  ( { x }  ~~  1o 
<->  { A }  ~~  1o ) )
3 vex 3116 . . 3  |-  x  e. 
_V
43ensn1 7580 . 2  |-  { x }  ~~  1o
52, 4vtoclg 3171 1  |-  ( A  e.  V  ->  { A }  ~~  1o )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    = wceq 1379    e. wcel 1767   {csn 4027   class class class wbr 4447   1oc1o 7124    ~~ cen 7514
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pr 4686  ax-un 6577
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-ral 2819  df-rex 2820  df-rab 2823  df-v 3115  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-nul 3786  df-if 3940  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-id 4795  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-1o 7131  df-en 7518
This theorem is referenced by:  enpr1g  7582  en1b  7584  en2sn  7596  snfi  7597  snnen2o  7707  sucxpdom  7730  en1eqsn  7750  en1eqsnbi  7751  pr2nelem  8383  prdom2  8385  cda1en  8556  rngosn4  25202  rngoueqz  25205  snct  27303
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