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Theorem en1 9064
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7755. (Revised by BTernaryTau, 23-Sep-2024.)
Assertion
Ref Expression
en1 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Distinct variable group:   𝑥,𝐴

Proof of Theorem en1
Dummy variable 𝑓 is distinct from all other variables.
StepHypRef Expression
1 df1o2 8513 . . . . 5 1o = {∅}
21breq2i 5151 . . . 4 (𝐴 ≈ 1o𝐴 ≈ {∅})
3 encv 8993 . . . . . 6 (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4 breng 8994 . . . . . 6 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅}))
53, 4syl 17 . . . . 5 (𝐴 ≈ {∅} → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅}))
65ibi 267 . . . 4 (𝐴 ≈ {∅} → ∃𝑓 𝑓:𝐴1-1-onto→{∅})
72, 6sylbi 217 . . 3 (𝐴 ≈ 1o → ∃𝑓 𝑓:𝐴1-1-onto→{∅})
8 f1ocnv 6860 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
9 f1ofo 6855 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
10 forn 6823 . . . . . . 7 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
119, 10syl 17 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
12 f1of 6848 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
13 0ex 5307 . . . . . . . . . . 11 ∅ ∈ V
1413fsn2 7156 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1514simprbi 496 . . . . . . . . 9 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1612, 15syl 17 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1716rneqd 5949 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1813rnsnop 6244 . . . . . . 7 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1917, 18eqtrdi 2793 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
2011, 19eqtr3d 2779 . . . . 5 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
21 fvex 6919 . . . . . 6 (𝑓‘∅) ∈ V
22 sneq 4636 . . . . . . 7 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2322eqeq2d 2748 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2421, 23spcev 3606 . . . . 5 (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
258, 20, 243syl 18 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2625exlimiv 1930 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
277, 26syl 17 . 2 (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28 vex 3484 . . . . 5 𝑥 ∈ V
2928ensn1 9061 . . . 4 {𝑥} ≈ 1o
30 breq1 5146 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
3129, 30mpbiri 258 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1o)
3231exlimiv 1930 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
3327, 32impbii 209 1 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 206  wa 395   = wceq 1540  wex 1779  wcel 2108  Vcvv 3480  c0 4333  {csn 4626  cop 4632   class class class wbr 5143  ccnv 5684  ran crn 5686  wf 6557  ontowfo 6559  1-1-ontowf1o 6560  cfv 6561  1oc1o 8499  cen 8982
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-10 2141  ax-12 2177  ax-ext 2708  ax-sep 5296  ax-nul 5306  ax-pr 5432
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2065  df-mo 2540  df-eu 2569  df-clab 2715  df-cleq 2729  df-clel 2816  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3381  df-rab 3437  df-v 3482  df-dif 3954  df-un 3956  df-in 3958  df-ss 3968  df-nul 4334  df-if 4526  df-sn 4627  df-pr 4629  df-op 4633  df-uni 4908  df-br 5144  df-opab 5206  df-id 5578  df-xp 5691  df-rel 5692  df-cnv 5693  df-co 5694  df-dm 5695  df-rn 5696  df-res 5697  df-ima 5698  df-suc 6390  df-iota 6514  df-fun 6563  df-fn 6564  df-f 6565  df-f1 6566  df-fo 6567  df-f1o 6568  df-fv 6569  df-1o 8506  df-en 8986
This theorem is referenced by:  en1b  9065  reuen1  9066  en1eqsn  9308  en2  9315  card1  10008  pm54.43  10041  hash1elsn  14410  hash1snb  14458  ufildom1  23934  unidifsnel  32553  unidifsnne  32554  funen1cnv  35102  lfuhgr3  35125  snen1g  43537  istermc3  49123
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