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Theorem ensn1g 9061
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 4641 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 5158 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3 vex 3482 . . 3 𝑥 ∈ V
43ensn1 9060 . 2 {𝑥} ≈ 1o
52, 4vtoclg 3554 1 (𝐴𝑉 → {𝐴} ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1537  wcel 2106  {csn 4631   class class class wbr 5148  1oc1o 8498  cen 8981
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-mo 2538  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-id 5583  df-xp 5695  df-rel 5696  df-cnv 5697  df-co 5698  df-dm 5699  df-rn 5700  df-suc 6392  df-fun 6565  df-fn 6566  df-f 6567  df-f1 6568  df-fo 6569  df-f1o 6570  df-1o 8505  df-en 8985
This theorem is referenced by:  enpr1g  9062  en1b  9064  snmapen1  9078  snfi  9082  snfiOLD  9083  enpr2dOLD  9089  snnen2oOLD  9262  sucxpdom  9289  en1eqsnOLD  9307  en1eqsnbi  9308  pr2nelemOLD  10041  prdom2  10044  dju1en  10210  triv1nsgd  19204  snct  32731  rngoueqz  37927  safesnsupfidom1o  43407  sn1dom  43516
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