Theorem ensn1g 8260

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 4378	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	breq1d 4853	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1𝑜 ↔ {𝐴} ≈ 1𝑜))
3		vex 3388	. . 3 ⊢ 𝑥 ∈ V
4	3	ensn1 8259	. 2 ⊢ {𝑥} ≈ 1𝑜
5	2, 4	vtoclg 3453	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1𝑜)

Syntax hints:   → wi 4   = wceq 1653   ∈ wcel 2157  {csn 4368   class class class wbr 4843  1_𝑜c1o 7792   ≈ cen 8192

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1891  ax-4 1905  ax-5 2006  ax-6 2072  ax-7 2107  ax-8 2159  ax-9 2166  ax-10 2185  ax-11 2200  ax-12 2213  ax-13 2377  ax-ext 2777  ax-sep 4975  ax-nul 4983  ax-pr 5097  ax-un 7183

This theorem depends on definitions:  df-bi 199  df-an 386  df-or 875  df-3an 1110  df-tru 1657  df-ex 1876  df-nf 1880  df-sb 2065  df-mo 2591  df-eu 2609  df-clab 2786  df-cleq 2792  df-clel 2795  df-nfc 2930  df-ral 3094  df-rex 3095  df-rab 3098  df-v 3387  df-dif 3772  df-un 3774  df-in 3776  df-ss 3783  df-nul 4116  df-if 4278  df-sn 4369  df-pr 4371  df-op 4375  df-uni 4629  df-br 4844  df-opab 4906  df-id 5220  df-xp 5318  df-rel 5319  df-cnv 5320  df-co 5321  df-dm 5322  df-rn 5323  df-suc 5947  df-fun 6103  df-fn 6104  df-f 6105  df-f1 6106  df-fo 6107  df-f1o 6108  df-1o 7799  df-en 8196

This theorem is referenced by:  enpr1g  8261  en1b  8263  snmapen1  8277  en2sn  8279  snfi  8280  snnen2o  8391  sucxpdom  8411  en1eqsn  8432  en1eqsnbi  8433  pr2nelem  9113  prdom2  9115  cda1en  9285  snct  30009  rngoueqz  34226