Theorem ensn1g 8962

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 4578	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	breq1d 5096	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3		vex 3434	. . 3 ⊢ 𝑥 ∈ V
4	3	ensn1 8961	. 2 ⊢ {𝑥} ≈ 1o
5	2, 4	vtoclg 3500	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)

Syntax hints:   → wi 4   = wceq 1542   ∈ wcel 2114  {csn 4568   class class class wbr 5086  1_oc1o 8391   ≈ cen 8883

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 2709  ax-sep 5231  ax-nul 5241  ax-pr 5370

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 2540  df-clab 2716  df-cleq 2729  df-clel 2812  df-ral 3053  df-rex 3063  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-id 5519  df-xp 5630  df-rel 5631  df-cnv 5632  df-co 5633  df-dm 5634  df-rn 5635  df-suc 6323  df-fun 6494  df-fn 6495  df-f 6496  df-f1 6497  df-fo 6498  df-f1o 6499  df-1o 8398  df-en 8887

This theorem is referenced by:  enpr1g  8963  en1b  8965  snmapen1  8979  snfi  8983  sucxpdom  9164  en1eqsnbi  9179  prdom2  9919  dju1en  10085  triv1nsgd  19139  snct  32800  rngoueqz  38275  safesnsupfidom1o  43862  sn1dom  43971