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Theorem ensn1 8964
Description: A singleton is equinumerous to ordinal one. (Contributed by NM, 4-Nov-2002.) Avoid ax-un 7673. (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 5389 . . . 4 {⟨𝐴, ∅⟩} ∈ V
2 f1oeq1 6773 . . . 4 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
3 ensn1.1 . . . . 5 𝐴 ∈ V
4 0ex 5265 . . . . 5 ∅ ∈ V
53, 4f1osn 6825 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
61, 2, 5ceqsexv2d 3496 . . 3 𝑓 𝑓:{𝐴}–1-1-onto→{∅}
7 snex 5389 . . . 4 {𝐴} ∈ V
8 snex 5389 . . . 4 {∅} ∈ V
9 breng 8895 . . . 4 (({𝐴} ∈ V ∧ {∅} ∈ V) → ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅}))
107, 8, 9mp2an 691 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
116, 10mpbir 230 . 2 {𝐴} ≈ {∅}
12 df1o2 8420 . 2 1o = {∅}
1311, 12breqtrri 5133 1 {𝐴} ≈ 1o
Colors of variables: wff setvar class
Syntax hints:  wb 205  wex 1782  wcel 2107  Vcvv 3444  c0 4283  {csn 4587  cop 4593   class class class wbr 5106  1-1-ontowf1o 6496  1oc1o 8406  cen 8883
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5257  ax-nul 5264  ax-pr 5385
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-mo 2535  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3062  df-rex 3071  df-rab 3407  df-v 3446  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4284  df-if 4488  df-sn 4588  df-pr 4590  df-op 4594  df-br 5107  df-opab 5169  df-id 5532  df-xp 5640  df-rel 5641  df-cnv 5642  df-co 5643  df-dm 5644  df-rn 5645  df-suc 6324  df-fun 6499  df-fn 6500  df-f 6501  df-f1 6502  df-fo 6503  df-f1o 6504  df-1o 8413  df-en 8887
This theorem is referenced by:  ensn1g  8966  en1  8968  en1OLD  8969  sdom1  9189  fodomfi  9272  pm54.43  9942  1nprm  16560  gex1  19378  sylow2a  19406  0frgp  19566  en1top  22350  en2top  22351  t1connperf  22803  ptcmplem2  23420  xrge0tsms2  24214  sconnpi1  33890
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