MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  ensn1 Structured version   Visualization version   GIF version

Theorem ensn1 7980
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 4760 . . . . 5 ∅ ∈ V
31, 2f1osn 6143 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
4 snex 4879 . . . . 5 {⟨𝐴, ∅⟩} ∈ V
5 f1oeq1 6094 . . . . 5 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
64, 5spcev 3290 . . . 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 7924 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
97, 8mpbir 221 . 2 {𝐴} ≈ {∅}
10 df1o2 7532 . 2 1𝑜 = {∅}
119, 10breqtrri 4650 1 {𝐴} ≈ 1𝑜
Colors of variables: wff setvar class
Syntax hints:  wex 1701  wcel 1987  Vcvv 3190  c0 3897  {csn 4155  cop 4161   class class class wbr 4623  1-1-ontowf1o 5856  1𝑜c1o 7513  cen 7912
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 4751  ax-nul 4759  ax-pr 4877  ax-un 6914
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 2913  df-rex 2914  df-rab 2917  df-v 3192  df-dif 3563  df-un 3565  df-in 3567  df-ss 3574  df-nul 3898  df-if 4065  df-sn 4156  df-pr 4158  df-op 4162  df-uni 4410  df-br 4624  df-opab 4684  df-id 4999  df-xp 5090  df-rel 5091  df-cnv 5092  df-co 5093  df-dm 5094  df-rn 5095  df-suc 5698  df-fun 5859  df-fn 5860  df-f 5861  df-f1 5862  df-fo 5863  df-f1o 5864  df-1o 7520  df-en 7916
This theorem is referenced by:  ensn1g  7981  en1  7983  fodomfi  8199  pm54.43  8786  1nprm  15335  isprm2lem  15337  gex1  17946  sylow2a  17974  0frgp  18132  en1top  20728  en2top  20729  t1connperf  21179  ptcmplem2  21797  xrge0tsms2  22578  sconnpi1  30982
  Copyright terms: Public domain W3C validator