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Theorem ensn1 7964
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.)
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 ensn1.1 . . . . 5 𝐴 ∈ V
2 0ex 4750 . . . . 5 ∅ ∈ V
31, 2f1osn 6133 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
4 snex 4869 . . . . 5 {⟨𝐴, ∅⟩} ∈ V
5 f1oeq1 6084 . . . . 5 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
64, 5spcev 3286 . . . 4 ({⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
73, 6ax-mp 5 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
8 bren 7908 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
97, 8mpbir 221 . 2 {𝐴} ≈ {∅}
10 df1o2 7517 . 2 1𝑜 = {∅}
119, 10breqtrri 4640 1 {𝐴} ≈ 1𝑜
Colors of variables: wff setvar class
Syntax hints:  wex 1701  wcel 1987  Vcvv 3186  c0 3891  {csn 4148  cop 4154   class class class wbr 4613  1-1-ontowf1o 5846  1𝑜c1o 7498  cen 7896
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1719  ax-4 1734  ax-5 1836  ax-6 1885  ax-7 1932  ax-8 1989  ax-9 1996  ax-10 2016  ax-11 2031  ax-12 2044  ax-13 2245  ax-ext 2601  ax-sep 4741  ax-nul 4749  ax-pr 4867  ax-un 6902
This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1038  df-tru 1483  df-ex 1702  df-nf 1707  df-sb 1878  df-eu 2473  df-mo 2474  df-clab 2608  df-cleq 2614  df-clel 2617  df-nfc 2750  df-ral 2912  df-rex 2913  df-rab 2916  df-v 3188  df-dif 3558  df-un 3560  df-in 3562  df-ss 3569  df-nul 3892  df-if 4059  df-sn 4149  df-pr 4151  df-op 4155  df-uni 4403  df-br 4614  df-opab 4674  df-id 4989  df-xp 5080  df-rel 5081  df-cnv 5082  df-co 5083  df-dm 5084  df-rn 5085  df-suc 5688  df-fun 5849  df-fn 5850  df-f 5851  df-f1 5852  df-fo 5853  df-f1o 5854  df-1o 7505  df-en 7900
This theorem is referenced by:  ensn1g  7965  en1  7967  fodomfi  8183  pm54.43  8770  1nprm  15316  isprm2lem  15318  gex1  17927  sylow2a  17955  0frgp  18113  en1top  20699  en2top  20700  t1connperf  21149  ptcmplem2  21767  xrge0tsms2  22546  sconnpi1  30929
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