Theorem ensn1 9061

Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7755. (Revised by BTernaryTau, 23-Sep-2024.)

Hypothesis

Ref	Expression
ensn1.1	⊢ 𝐴 ∈ V

Assertion

Ref	Expression
ensn1	⊢ {𝐴} ≈ 1o

Proof of Theorem ensn1

Dummy variable 𝑓 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		snex 5436	. . . 4 ⊢ {⟨𝐴, ∅⟩} ∈ V
2		f1oeq1 6836	. . . 4 ⊢ (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3		ensn1.1	. . . . 5 ⊢ 𝐴 ∈ V
4		0ex 5307	. . . . 5 ⊢ ∅ ∈ V
5	3, 4	f1osn 6888	. . . 4 ⊢ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
6	1, 2, 5	ceqsexv2d 3533	. . 3 ⊢ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7		snex 5436	. . . 4 ⊢ {𝐴} ∈ V
8		snex 5436	. . . 4 ⊢ {∅} ∈ V
9		breng 8994	. . . 4 ⊢ (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}))
10	7, 8, 9	mp2an 692	. . 3 ⊢ ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
11	6, 10	mpbir 231	. 2 ⊢ {𝐴} ≈ {∅}
12		df1o2 8513	. 2 ⊢ 1o = {∅}
13	11, 12	breqtrri 5170	1 ⊢ {𝐴} ≈ 1o

Syntax hints:   ↔ wb 206  ∃wex 1779   ∈ wcel 2108  Vcvv 3480  ∅c0 4333  {csn 4626  ⟨cop 4632   class class class wbr 5143  –1-1-onto→wf1o 6560  1_oc1o 8499   ≈ cen 8982

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 2007  ax-8 2110  ax-9 2118  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-mo 2540  df-clab 2715  df-cleq 2729  df-clel 2816  df-ral 3062  df-rex 3071  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-suc 6390  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-1o 8506  df-en 8986

This theorem is referenced by:  ensn1g  9062  en1  9064  sdom1  9278  fodomfiOLD  9370  pm54.43  10041  1nprm  16716  gex1  19609  sylow2a  19637  0frgp  19797  en1top  22991  en2top  22992  t1connperf  23444  ptcmplem2  24061  xrge0tsms2  24857  sconnpi1  35244  setcsnterm  49133