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Theorem ensn1 7906
 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 4718 . . . . 5 ∅ ∈ V
31, 2f1osn 6088 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
4 snex 4835 . . . . 5 {⟨𝐴, ∅⟩} ∈ V
5 f1oeq1 6040 . . . . 5 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
64, 5spcev 3273 . . . 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 7850 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
97, 8mpbir 220 . 2 {𝐴} ≈ {∅}
10 df1o2 7459 . 2 1𝑜 = {∅}
119, 10breqtrri 4610 1 {𝐴} ≈ 1𝑜
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1695   ∈ wcel 1977  Vcvv 3173  ∅c0 3874  {csn 4125  ⟨cop 4131   class class class wbr 4583  –1-1-onto→wf1o 5803  1𝑜c1o 7440   ≈ cen 7838 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-8 1979  ax-9 1986  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590  ax-sep 4709  ax-nul 4717  ax-pr 4833  ax-un 6847 This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-eu 2462  df-mo 2463  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rex 2902  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-uni 4373  df-br 4584  df-opab 4644  df-id 4953  df-xp 5044  df-rel 5045  df-cnv 5046  df-co 5047  df-dm 5048  df-rn 5049  df-suc 5646  df-fun 5806  df-fn 5807  df-f 5808  df-f1 5809  df-fo 5810  df-f1o 5811  df-1o 7447  df-en 7842 This theorem is referenced by:  ensn1g  7907  en1  7909  fodomfi  8124  pm54.43  8709  1nprm  15230  isprm2lem  15232  gex1  17829  sylow2a  17857  0frgp  18015  en1top  20599  en2top  20600  t1conperf  21049  ptcmplem2  21667  xrge0tsms2  22446  sconpi1  30475
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