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Theorem ensn1g 7966
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)
Assertion
Ref Expression
ensn1g (𝐴𝑉 → {𝐴} ≈ 1𝑜)

Proof of Theorem ensn1g
Dummy variable 𝑥 is distinct from all other variables.
StepHypRef Expression
1 sneq 4163 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 4628 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1𝑜 ↔ {𝐴} ≈ 1𝑜))
3 vex 3194 . . 3 𝑥 ∈ V
43ensn1 7965 . 2 {𝑥} ≈ 1𝑜
52, 4vtoclg 3257 1 (𝐴𝑉 → {𝐴} ≈ 1𝑜)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1480  wcel 1992  {csn 4153   class class class wbr 4618  1𝑜c1o 7499  cen 7897
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1841  ax-6 1890  ax-7 1937  ax-8 1994  ax-9 2001  ax-10 2021  ax-11 2036  ax-12 2049  ax-13 2250  ax-ext 2606  ax-sep 4746  ax-nul 4754  ax-pr 4872  ax-un 6903
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1883  df-eu 2478  df-mo 2479  df-clab 2613  df-cleq 2619  df-clel 2622  df-nfc 2756  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3193  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3897  df-if 4064  df-sn 4154  df-pr 4156  df-op 4160  df-uni 4408  df-br 4619  df-opab 4679  df-id 4994  df-xp 5085  df-rel 5086  df-cnv 5087  df-co 5088  df-dm 5089  df-rn 5090  df-suc 5691  df-fun 5852  df-fn 5853  df-f 5854  df-f1 5855  df-fo 5856  df-f1o 5857  df-1o 7506  df-en 7901
This theorem is referenced by:  enpr1g  7967  en1b  7969  en2sn  7982  snfi  7983  snnen2o  8094  sucxpdom  8114  en1eqsn  8135  en1eqsnbi  8136  pr2nelem  8772  prdom2  8774  cda1en  8942  snct  29325  rngoueqz  33357
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