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Theorem ensn1 8187
 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 4942 . . . . 5 ∅ ∈ V
31, 2f1osn 6338 . . . 4 {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}
4 snex 5057 . . . . 5 {⟨𝐴, ∅⟩} ∈ V
5 f1oeq1 6289 . . . . 5 (𝑓 = {⟨𝐴, ∅⟩} → (𝑓:{𝐴}–1-1-onto→{∅} ↔ {⟨𝐴, ∅⟩}:{𝐴}–1-1-onto→{∅}))
64, 5spcev 3440 . . . 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 8132 . . 3 ({𝐴} ≈ {∅} ↔ ∃𝑓 𝑓:{𝐴}–1-1-onto→{∅})
97, 8mpbir 221 . 2 {𝐴} ≈ {∅}
10 df1o2 7743 . 2 1𝑜 = {∅}
119, 10breqtrri 4831 1 {𝐴} ≈ 1𝑜
 Colors of variables: wff setvar class Syntax hints:  ∃wex 1853   ∈ wcel 2139  Vcvv 3340  ∅c0 4058  {csn 4321  ⟨cop 4327   class class class wbr 4804  –1-1-onto→wf1o 6048  1𝑜c1o 7723   ≈ cen 8120 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055  ax-un 7115 This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-uni 4589  df-br 4805  df-opab 4865  df-id 5174  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-suc 5890  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-1o 7730  df-en 8124 This theorem is referenced by:  ensn1g  8188  en1  8190  fodomfi  8406  pm54.43  9036  1nprm  15614  gex1  18226  sylow2a  18254  0frgp  18412  en1top  21010  en2top  21011  t1connperf  21461  ptcmplem2  22078  xrge0tsms2  22859  sconnpi1  31549
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