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Theorem en1 8972
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7677. (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 8424 . . . . 5 1o = {∅}
21breq2i 5118 . . . 4 (𝐴 ≈ 1o𝐴 ≈ {∅})
3 encv 8898 . . . . . 6 (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4 breng 8899 . . . . . 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 216 . . 3 (𝐴 ≈ 1o → ∃𝑓 𝑓:𝐴1-1-onto→{∅})
8 f1ocnv 6801 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
9 f1ofo 6796 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
10 forn 6764 . . . . . . 7 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
119, 10syl 17 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
12 f1of 6789 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
13 0ex 5269 . . . . . . . . . . 11 ∅ ∈ V
1413fsn2 7087 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1514simprbi 498 . . . . . . . . 9 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1612, 15syl 17 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1716rneqd 5898 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1813rnsnop 6181 . . . . . . 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 6860 . . . . . 6 (𝑓‘∅) ∈ V
22 sneq 4601 . . . . . . 7 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2322eqeq2d 2748 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2421, 23spcev 3568 . . . . 5 (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
258, 20, 243syl 18 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2625exlimiv 1934 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
277, 26syl 17 . 2 (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28 vex 3452 . . . . 5 𝑥 ∈ V
2928ensn1 8968 . . . 4 {𝑥} ≈ 1o
30 breq1 5113 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
3129, 30mpbiri 258 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1o)
3231exlimiv 1934 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1o)
3327, 32impbii 208 1 (𝐴 ≈ 1o ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff setvar class
Syntax hints:  wb 205  wa 397   = wceq 1542  wex 1782  wcel 2107  Vcvv 3448  c0 4287  {csn 4591  cop 4597   class class class wbr 5110  ccnv 5637  ran crn 5639  wf 6497  ontowfo 6499  1-1-ontowf1o 6500  cfv 6501  1oc1o 8410  cen 8887
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-10 2138  ax-12 2172  ax-ext 2708  ax-sep 5261  ax-nul 5268  ax-pr 5389
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-nf 1787  df-sb 2069  df-mo 2539  df-eu 2568  df-clab 2715  df-cleq 2729  df-clel 2815  df-ne 2945  df-ral 3066  df-rex 3075  df-reu 3357  df-rab 3411  df-v 3450  df-dif 3918  df-un 3920  df-in 3922  df-ss 3932  df-nul 4288  df-if 4492  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4871  df-br 5111  df-opab 5173  df-id 5536  df-xp 5644  df-rel 5645  df-cnv 5646  df-co 5647  df-dm 5648  df-rn 5649  df-res 5650  df-ima 5651  df-suc 6328  df-iota 6453  df-fun 6503  df-fn 6504  df-f 6505  df-f1 6506  df-fo 6507  df-f1o 6508  df-fv 6509  df-1o 8417  df-en 8891
This theorem is referenced by:  en1b  8974  en1bOLD  8975  reuen1  8976  en1eqsn  9225  en2  9232  card1  9911  pm54.43  9944  hash1elsn  14278  hash1snb  14326  ufildom1  23293  unidifsnel  31504  unidifsnne  31505  funen1cnv  33732  lfuhgr3  33753  snen1g  41870
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