Theorem ensn1g 9015

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 4601	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	breq1d 5120	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3		vex 3467	. . 3 ⊢ 𝑥 ∈ V
4	3	ensn1 9014	. 2 ⊢ {𝑥} ≈ 1o
5	2, 4	vtoclg 3531	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)

Syntax hints:   → wi 4   = wceq 1567   ∈ wcel 2149  {csn 4591   class class class wbr 5110  1_oc1o 8442   ≈ cen 8936

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1822  ax-4 1836  ax-5 1937  ax-6 1994  ax-7 2035  ax-8 2151  ax-9 2159  ax-ext 2741  ax-sep 5258  ax-nul 5268  ax-pr 5402

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1570  df-fal 1580  df-ex 1807  df-sb 2098  df-mo 2573  df-clab 2748  df-cleq 2761  df-clel 2844  df-ral 3086  df-rex 3096  df-rab 3424  df-v 3465  df-dif 3916  df-un 3918  df-in 3920  df-ss 3930  df-nul 4295  df-if 4490  df-sn 4592  df-pr 4594  df-op 4598  df-br 5111  df-opab 5175  df-id 5554  df-xp 5665  df-rel 5666  df-cnv 5667  df-co 5668  df-dm 5669  df-rn 5670  df-suc 6364  df-fun 6536  df-fn 6537  df-f 6538  df-f1 6539  df-fo 6540  df-f1o 6541  df-1o 8449  df-en 8940

This theorem is referenced by:  enpr1g  9016  en1b  9018  snmapen1  9032  snfi  9036  sucxpdom  9217  en1eqsnbi  9232  prdom2  9986  dju1en  10151  triv1nsgd  19235  snct  32994  dflring3  33728  dflring4  33729  rngoueqz  38474  safesnsupfidom1o  44030  sn1dom  44139