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Theorem ensn1g 7131
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 3785 . . 3  |-  ( x  =  A  ->  { x }  =  { A } )
21breq1d 4182 . 2  |-  ( x  =  A  ->  ( { x }  ~~  1o 
<->  { A }  ~~  1o ) )
3 vex 2919 . . 3  |-  x  e. 
_V
43ensn1 7130 . 2  |-  { x }  ~~  1o
52, 4vtoclg 2971 1  |-  ( A  e.  V  ->  { A }  ~~  1o )
Colors of variables: wff set class
Syntax hints:    -> wi 4    = wceq 1649    e. wcel 1721   {csn 3774   class class class wbr 4172   1oc1o 6676    ~~ cen 7065
This theorem is referenced by:  enpr1g  7132  en1b  7134  en2sn  7145  snfi  7146  sucxpdom  7277  en1eqsn  7297  pr2nelem  7844  prdom2  7846  cda1en  8011  rngosn4  21968  rngoueqz  21971  snct  24056
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-3 7  ax-mp 8  ax-gen 1552  ax-5 1563  ax-17 1623  ax-9 1662  ax-8 1683  ax-13 1723  ax-14 1725  ax-6 1740  ax-7 1745  ax-11 1757  ax-12 1946  ax-ext 2385  ax-sep 4290  ax-nul 4298  ax-pr 4363  ax-un 4660
This theorem depends on definitions:  df-bi 178  df-or 360  df-an 361  df-3an 938  df-tru 1325  df-ex 1548  df-nf 1551  df-sb 1656  df-eu 2258  df-mo 2259  df-clab 2391  df-cleq 2397  df-clel 2400  df-nfc 2529  df-ne 2569  df-ral 2671  df-rex 2672  df-rab 2675  df-v 2918  df-dif 3283  df-un 3285  df-in 3287  df-ss 3294  df-nul 3589  df-if 3700  df-sn 3780  df-pr 3781  df-op 3783  df-uni 3976  df-br 4173  df-opab 4227  df-id 4458  df-suc 4547  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-1o 6683  df-en 7069
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