Theorem ensn1g 8062

Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 23-Apr-2004.)

Assertion

Ref	Expression
ensn1g	⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜)

Proof of Theorem ensn1g

Dummy variable 𝑥 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		sneq 4220	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	breq1d 4695	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1𝑜 ↔ {𝐴} ≈ 1𝑜))
3		vex 3234	. . 3 ⊢ 𝑥 ∈ V
4	3	ensn1 8061	. 2 ⊢ {𝑥} ≈ 1𝑜
5	2, 4	vtoclg 3297	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜)

Syntax hints:   → wi 4   = wceq 1523   ∈ wcel 2030  {csn 4210   class class class wbr 4685  1_𝑜c1o 7598   ≈ cen 7994

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-8 2032  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936  ax-un 6991

This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-eu 2502  df-mo 2503  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-uni 4469  df-br 4686  df-opab 4746  df-id 5053  df-xp 5149  df-rel 5150  df-cnv 5151  df-co 5152  df-dm 5153  df-rn 5154  df-suc 5767  df-fun 5928  df-fn 5929  df-f 5930  df-f1 5931  df-fo 5932  df-f1o 5933  df-1o 7605  df-en 7998

This theorem is referenced by:  enpr1g  8063  en1b  8065  en2sn  8078  snfi  8079  snnen2o  8190  sucxpdom  8210  en1eqsn  8231  en1eqsnbi  8232  pr2nelem  8865  prdom2  8867  cda1en  9035  snct  29619  rngoueqz  33869