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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.
StepHypRef 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
53, 4f1osn 6888 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 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→{∅}))
107, 8, 9mp2an 692 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
116, 10mpbir 231 . 2 {𝐴} ≈ {∅}
12 df1o2 8513 . 2 1o = {∅}
1311, 12breqtrri 5170 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wb 206  wex 1779  wcel 2108  Vcvv 3480  c0 4333  {csn 4626  cop 4632   class class class wbr 5143  1-1-ontowf1o 6560  1oc1o 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
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