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Theorem ensn1 9017
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7725. (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 5432 . . . 4 {⟨𝐴, ∅⟩} ∈ V
2 f1oeq1 6822 . . . 4 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3 ensn1.1 . . . . 5 𝐴 ∈ V
4 0ex 5308 . . . . 5 ∅ ∈ V
53, 4f1osn 6874 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 3529 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7 snex 5432 . . . 4 {𝐴} ∈ V
8 snex 5432 . . . 4 {∅} ∈ V
9 breng 8948 . . . 4 (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}))
107, 8, 9mp2an 691 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
116, 10mpbir 230 . 2 {𝐴} ≈ {∅}
12 df1o2 8473 . 2 1o = {∅}
1311, 12breqtrri 5176 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1782  wcel 2107  Vcvv 3475  c0 4323  {csn 4629  cop 4635   class class class wbr 5149  1-1-ontowf1o 6543  1oc1o 8459  cen 8936
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-id 5575  df-xp 5683  df-rel 5684  df-cnv 5685  df-co 5686  df-dm 5687  df-rn 5688  df-suc 6371  df-fun 6546  df-fn 6547  df-f 6548  df-f1 6549  df-fo 6550  df-f1o 6551  df-1o 8466  df-en 8940
This theorem is referenced by:  ensn1g  9019  en1  9021  en1OLD  9022  sdom1  9242  fodomfi  9325  pm54.43  9996  1nprm  16616  gex1  19459  sylow2a  19487  0frgp  19647  en1top  22487  en2top  22488  t1connperf  22940  ptcmplem2  23557  xrge0tsms2  24351  sconnpi1  34230
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