Theorem ensn1g 8996

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 4602	. . 3 ⊢ (𝑥 = 𝐴 → {𝑥} = {𝐴})
2	1	breq1d 5120	. 2 ⊢ (𝑥 = 𝐴 → ({𝑥} ≈ 1o ↔ {𝐴} ≈ 1o))
3		vex 3454	. . 3 ⊢ 𝑥 ∈ V
4	3	ensn1 8995	. 2 ⊢ {𝑥} ≈ 1o
5	2, 4	vtoclg 3523	1 ⊢ (𝐴 ∈ 𝑉 → {𝐴} ≈ 1o)

Syntax hints:   → wi 4   = wceq 1540   ∈ wcel 2109  {csn 4592   class class class wbr 5110  1_oc1o 8430   ≈ cen 8918

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 2702  ax-sep 5254  ax-nul 5264  ax-pr 5390

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 2534  df-clab 2709  df-cleq 2722  df-clel 2804  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3920  df-un 3922  df-ss 3934  df-nul 4300  df-if 4492  df-sn 4593  df-pr 4595  df-op 4599  df-br 5111  df-opab 5173  df-id 5536  df-xp 5647  df-rel 5648  df-cnv 5649  df-co 5650  df-dm 5651  df-rn 5652  df-suc 6341  df-fun 6516  df-fn 6517  df-f 6518  df-f1 6519  df-fo 6520  df-f1o 6521  df-1o 8437  df-en 8922

This theorem is referenced by:  enpr1g  8997  en1b  8999  snmapen1  9013  snfi  9017  snfiOLD  9018  enpr2dOLD  9024  sucxpdom  9209  en1eqsnOLD  9227  en1eqsnbi  9228  pr2nelemOLD  9963  prdom2  9966  dju1en  10132  triv1nsgd  19112  snct  32644  rngoueqz  37941  safesnsupfidom1o  43413  sn1dom  43522