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Theorem ensn1g 8563
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 4574 . . 3 (𝑥 = 𝐴 → {𝑥} = {𝐴})
21breq1d 5073 . 2 (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3 vex 3503 . . 3 𝑥 ∈ V
43ensn1 8562 . 2 {𝑥} ≈ 1o
52, 4vtoclg 3573 1 (𝐴𝑉 → {𝐴} ≈ 1o)
Colors of variables: wff setvar class
Syntax hints:  wi 4   = wceq 1530  wcel 2107  {csn 4564   class class class wbr 5063  1oc1o 8086  cen 8495
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798  ax-sep 5200  ax-nul 5207  ax-pr 5326  ax-un 7451
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-mo 2620  df-eu 2652  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rex 3149  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-uni 4838  df-br 5064  df-opab 5126  df-id 5459  df-xp 5560  df-rel 5561  df-cnv 5562  df-co 5563  df-dm 5564  df-rn 5565  df-suc 6195  df-fun 6354  df-fn 6355  df-f 6356  df-f1 6357  df-fo 6358  df-f1o 6359  df-1o 8093  df-en 8499
This theorem is referenced by:  enpr1g  8564  en1b  8566  snmapen1  8580  en2sn  8582  snfi  8583  enpr2d  8586  snnen2o  8696  sucxpdom  8716  en1eqsn  8737  en1eqsnbi  8738  pr2nelem  9419  prdom2  9421  dju1en  9586  triv1nsgd  18255  snct  30362  rngoueqz  35086  sn1dom  39757
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