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Theorem ensn1 9013
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7721. (Revised by BTernaryTau, 23-Sep-2024.)
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 5430 . . . 4 {⟨𝐴, ∅⟩} ∈ V
2 f1oeq1 6818 . . . 4 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3 ensn1.1 . . . . 5 𝐴 ∈ V
4 0ex 5306 . . . . 5 ∅ ∈ V
53, 4f1osn 6870 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 3528 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7 snex 5430 . . . 4 {𝐴} ∈ V
8 snex 5430 . . . 4 {∅} ∈ V
9 breng 8944 . . . 4 (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}))
107, 8, 9mp2an 690 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
116, 10mpbir 230 . 2 {𝐴} ≈ {∅}
12 df1o2 8469 . 2 1o = {∅}
1311, 12breqtrri 5174 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1781  wcel 2106  Vcvv 3474  c0 4321  {csn 4627  cop 4633   class class class wbr 5147  1-1-ontowf1o 6539  1oc1o 8455  cen 8932
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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703  ax-sep 5298  ax-nul 5305  ax-pr 5426
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-mo 2534  df-clab 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rex 3071  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-opab 5210  df-id 5573  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-suc 6367  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-1o 8462  df-en 8936
This theorem is referenced by:  ensn1g  9015  en1  9017  en1OLD  9018  sdom1  9238  fodomfi  9321  pm54.43  9992  1nprm  16612  gex1  19453  sylow2a  19481  0frgp  19641  en1top  22478  en2top  22479  t1connperf  22931  ptcmplem2  23548  xrge0tsms2  24342  sconnpi1  34218
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