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Theorem ensn1 8563
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 snex 5319 . . . 4 {⟨𝐴, ∅⟩} ∈ V
2 f1oeq1 6592 . . . 4 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3 ensn1.1 . . . . 5 𝐴 ∈ V
4 0ex 5197 . . . . 5 ∅ ∈ V
53, 4f1osn 6642 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 3528 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7 bren 8508 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
86, 7mpbir 234 . 2 {𝐴} ≈ {∅}
9 df1o2 8106 . 2 1o = {∅}
108, 9breqtrri 5079 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wex 1781  wcel 2115  Vcvv 3480  c0 4275  {csn 4549  cop 4555   class class class wbr 5052  1-1-ontowf1o 6342  1oc1o 8085  cen 8496
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5189  ax-nul 5196  ax-pr 5317  ax-un 7451
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ral 3138  df-rex 3139  df-rab 3142  df-v 3482  df-dif 3922  df-un 3924  df-in 3926  df-ss 3936  df-nul 4276  df-if 4450  df-sn 4550  df-pr 4552  df-op 4556  df-uni 4825  df-br 5053  df-opab 5115  df-id 5447  df-xp 5548  df-rel 5549  df-cnv 5550  df-co 5551  df-dm 5552  df-rn 5553  df-suc 6184  df-fun 6345  df-fn 6346  df-f 6347  df-f1 6348  df-fo 6349  df-f1o 6350  df-1o 8092  df-en 8500
This theorem is referenced by:  ensn1g  8564  en1  8566  fodomfi  8788  pm54.43  9421  1nprm  16017  gex1  18712  sylow2a  18740  0frgp  18901  en1top  21585  en2top  21586  t1connperf  22037  ptcmplem2  22654  xrge0tsms2  23436  sconnpi1  32511
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