Theorem ensn1g 8021

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.

Step	Hyp	Ref	Expression
1		sneq 4187	. . 3 |- ( x = A -> { x } = { A } )
2	1	breq1d 4663	. 2 |- ( x = A -> ( { x } ~~ 1o <-> { A } ~~ 1o ) )
3		vex 3203	. . 3 |- x e. _V
4	3	ensn1 8020	. 2 |- { x } ~~ 1o
5	2, 4	vtoclg 3266	1 |- ( A e. V -> { A } ~~ 1o )

Syntax hints:   -> wi 4   = wceq 1483   e. wcel 1990   {csn 4177   class class class wbr 4653   1oc1o 7553   ~~ cen 7952

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-8 1992  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906  ax-un 6949

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-id 5024  df-xp 5120  df-rel 5121  df-cnv 5122  df-co 5123  df-dm 5124  df-rn 5125  df-suc 5729  df-fun 5890  df-fn 5891  df-f 5892  df-f1 5893  df-fo 5894  df-f1o 5895  df-1o 7560  df-en 7956

This theorem is referenced by:  enpr1g  8022  en1b  8024  en2sn  8037  snfi  8038  snnen2o  8149  sucxpdom  8169  en1eqsn  8190  en1eqsnbi  8191  pr2nelem  8827  prdom2  8829  cda1en  8997  snct  29491  rngoueqz  33739