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Theorem ensn1 6212
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 3875 . . . . 5 V
31, 2f1osn 5109 . . . 4 {⟨A, ∅⟩}:{A}–1-1-onto→{∅}
41, 2opex 3957 . . . . . 6 A, ∅⟩ V
54snex 3928 . . . . 5 {⟨A, ∅⟩} V
6 f1oeq1 5060 . . . . 5 (f = {⟨A, ∅⟩} → (f:{A}–1-1-onto→{∅} ↔ {⟨A, ∅⟩}:{A}–1-1-onto→{∅}))
75, 6spcev 2641 . . . 4 ({⟨A, ∅⟩}:{A}–1-1-onto→{∅} → f f:{A}–1-1-onto→{∅})
83, 7ax-mp 7 . . 3 f f:{A}–1-1-onto→{∅}
9 bren 6164 . . 3 ({A} ≈ {∅} ↔ f f:{A}–1-1-onto→{∅})
108, 9mpbir 134 . 2 {A} ≈ {∅}
11 df1o2 5952 . 2 1𝑜 = {∅}
1210, 11breqtrri 3780 1 {A} ≈ 1𝑜
Colors of variables: wff set class
Syntax hints:  wex 1378   wcel 1390  Vcvv 2551  c0 3218  {csn 3367  cop 3370   class class class wbr 3755  1-1-ontowf1o 4844  1𝑜c1o 5933  cen 6155
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-13 1401  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-nul 3874  ax-pow 3918  ax-pr 3935  ax-un 4136
This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-dif 2914  df-un 2916  df-in 2918  df-ss 2925  df-nul 3219  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-uni 3572  df-br 3756  df-opab 3810  df-id 4021  df-suc 4074  df-xp 4294  df-rel 4295  df-cnv 4296  df-co 4297  df-dm 4298  df-rn 4299  df-fun 4847  df-fn 4848  df-f 4849  df-f1 4850  df-fo 4851  df-f1o 4852  df-1o 5940  df-en 6158
This theorem is referenced by:  ensn1g  6213  en1  6215
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