Theorem ensn1g 8954

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 4589	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	breq1d 5105	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3		vex 3442	. . 3 ⊢ 𝑥 ∈ V
4	3	ensn1 8953	. 2 ⊢ {𝑥} ≈ 1o
5	2, 4	vtoclg 3511	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4579   class class class wbr 5095  1_oc1o 8388   ≈ cen 8876

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701  ax-sep 5238  ax-nul 5248  ax-pr 5374

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-mo 2533  df-clab 2708  df-cleq 2721  df-clel 2803  df-ral 3045  df-rex 3054  df-rab 3397  df-v 3440  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4479  df-sn 4580  df-pr 4582  df-op 4586  df-br 5096  df-opab 5158  df-id 5518  df-xp 5629  df-rel 5630  df-cnv 5631  df-co 5632  df-dm 5633  df-rn 5634  df-suc 6317  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-1o 8395  df-en 8880

This theorem is referenced by:  enpr1g  8955  en1b  8957  snmapen1  8971  snfi  8975  snfiOLD  8976  sucxpdom  9160  en1eqsnOLD  9178  en1eqsnbi  9179  prdom2  9919  dju1en  10085  triv1nsgd  19070  snct  32670  rngoueqz  37922  safesnsupfidom1o  43393  sn1dom  43502