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Theorem ensn1g 7175
 Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
Assertion
Ref Expression
ensn1g

Proof of Theorem ensn1g
Dummy variable is distinct from all other variables.
StepHypRef Expression
1 sneq 3827 . . 3
21breq1d 4225 . 2
3 vex 2961 . . 3
43ensn1 7174 . 2
52, 4vtoclg 3013 1
 Colors of variables: wff set class Syntax hints:   wi 4   wceq 1653   wcel 1726  csn 3816   class class class wbr 4215  c1o 6720   cen 7109 This theorem is referenced by:  enpr1g  7176  en1b  7178  en2sn  7189  snfi  7190  sucxpdom  7321  en1eqsn  7341  pr2nelem  7893  prdom2  7895  cda1en  8060  rngosn4  22020  rngoueqz  22023  snct  24108 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1556  ax-5 1567  ax-17 1627  ax-9 1667  ax-8 1688  ax-13 1728  ax-14 1730  ax-6 1745  ax-7 1750  ax-11 1762  ax-12 1951  ax-ext 2419  ax-sep 4333  ax-nul 4341  ax-pr 4406  ax-un 4704 This theorem depends on definitions:  df-bi 179  df-or 361  df-an 362  df-3an 939  df-tru 1329  df-ex 1552  df-nf 1555  df-sb 1660  df-eu 2287  df-mo 2288  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-ral 2712  df-rex 2713  df-rab 2716  df-v 2960  df-dif 3325  df-un 3327  df-in 3329  df-ss 3336  df-nul 3631  df-if 3742  df-sn 3822  df-pr 3823  df-op 3825  df-uni 4018  df-br 4216  df-opab 4270  df-id 4501  df-suc 4590  df-xp 4887  df-rel 4888  df-cnv 4889  df-co 4890  df-dm 4891  df-rn 4892  df-fun 5459  df-fn 5460  df-f 5461  df-f1 5462  df-fo 5463  df-f1o 5464  df-1o 6727  df-en 7113
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