Theorem ensn1g 8969

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 4577	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	breq1d 5095	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3		vex 3433	. . 3 ⊢ 𝑥 ∈ V
4	3	ensn1 8968	. 2 ⊢ {𝑥} ≈ 1o
5	2, 4	vtoclg 3499	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4567   class class class wbr 5085  1_oc1o 8398   ≈ cen 8890

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708  ax-sep 5231  ax-nul 5241  ax-pr 5375

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-mo 2539  df-clab 2715  df-cleq 2728  df-clel 2811  df-ral 3052  df-rex 3062  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-in 3896  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-id 5526  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-suc 6329  df-fun 6500  df-fn 6501  df-f 6502  df-f1 6503  df-fo 6504  df-f1o 6505  df-1o 8405  df-en 8894

This theorem is referenced by:  enpr1g  8970  en1b  8972  snmapen1  8986  snfi  8990  sucxpdom  9171  en1eqsnbi  9186  prdom2  9928  dju1en  10094  triv1nsgd  19148  snct  32785  rngoueqz  38261  safesnsupfidom1o  43844  sn1dom  43953