ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ensn1 GIF version

Theorem ensn1 6658
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 4025 . . . . 5 ∅ ∈ V
31, 2f1osn 5375 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
41, 2opex 4121 . . . . . 6 𝐴, ∅⟩ ∈ V
54snex 4079 . . . . 5 {⟨𝐴, ∅⟩} ∈ V
6 f1oeq1 5326 . . . . 5 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
75, 6spcev 2754 . . . 4 ({⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅} → ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
83, 7ax-mp 5 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
9 bren 6609 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
108, 9mpbir 145 . 2 {𝐴} ≈ {∅}
11 df1o2 6294 . 2 1o = {∅}
1210, 11breqtrri 3925 1 {𝐴} ≈ 1o
Colors of variables: wff set class
Syntax hints:  wex 1453  wcel 1465  Vcvv 2660  c0 3333  {csn 3497  cop 3500   class class class wbr 3899  1-1-ontowf1o 5092  1oc1o 6274  cen 6600
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 588  ax-in2 589  ax-io 683  ax-5 1408  ax-7 1409  ax-gen 1410  ax-ie1 1454  ax-ie2 1455  ax-8 1467  ax-10 1468  ax-11 1469  ax-i12 1470  ax-bndl 1471  ax-4 1472  ax-13 1476  ax-14 1477  ax-17 1491  ax-i9 1495  ax-ial 1499  ax-i5r 1500  ax-ext 2099  ax-sep 4016  ax-nul 4024  ax-pow 4068  ax-pr 4101  ax-un 4325
This theorem depends on definitions:  df-bi 116  df-3an 949  df-tru 1319  df-nf 1422  df-sb 1721  df-eu 1980  df-mo 1981  df-clab 2104  df-cleq 2110  df-clel 2113  df-nfc 2247  df-ral 2398  df-rex 2399  df-v 2662  df-dif 3043  df-un 3045  df-in 3047  df-ss 3054  df-nul 3334  df-pw 3482  df-sn 3503  df-pr 3504  df-op 3506  df-uni 3707  df-br 3900  df-opab 3960  df-id 4185  df-suc 4263  df-xp 4515  df-rel 4516  df-cnv 4517  df-co 4518  df-dm 4519  df-rn 4520  df-fun 5095  df-fn 5096  df-f 5097  df-f1 5098  df-fo 5099  df-f1o 5100  df-1o 6281  df-en 6603
This theorem is referenced by:  ensn1g  6659  en1  6661  pm54.43  7014  1nprm  11722  en1top  12173
  Copyright terms: Public domain W3C validator