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Theorem en1 9017
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.) Avoid ax-un 7721. (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 8469 . . . . 5 1o = {∅}
21breq2i 5155 . . . 4 (𝐴 ≈ 1o𝐴 ≈ {∅})
3 encv 8943 . . . . . 6 (𝐴 ≈ {∅} → (𝐴 ∈ V ∧ {∅} ∈ V))
4 breng 8944 . . . . . 6 ((𝐴 ∈ V ∧ {∅} ∈ V) → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅}))
53, 4syl 17 . . . . 5 (𝐴 ≈ {∅} → (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅}))
65ibi 266 . . . 4 (𝐴 ≈ {∅} → ∃𝑓 𝑓:𝐴1-1-onto→{∅})
72, 6sylbi 216 . . 3 (𝐴 ≈ 1o → ∃𝑓 𝑓:𝐴1-1-onto→{∅})
8 f1ocnv 6842 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
9 f1ofo 6837 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
10 forn 6805 . . . . . . 7 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
119, 10syl 17 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
12 f1of 6830 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
13 0ex 5306 . . . . . . . . . . 11 ∅ ∈ V
1413fsn2 7130 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1514simprbi 497 . . . . . . . . 9 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1612, 15syl 17 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1716rneqd 5935 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1813rnsnop 6220 . . . . . . 7 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1917, 18eqtrdi 2788 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
2011, 19eqtr3d 2774 . . . . 5 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
21 fvex 6901 . . . . . 6 (𝑓‘∅) ∈ V
22 sneq 4637 . . . . . . 7 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2322eqeq2d 2743 . . . . . 6 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2421, 23spcev 3596 . . . . 5 (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥})
258, 20, 243syl 18 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2625exlimiv 1933 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
277, 26syl 17 . 2 (𝐴 ≈ 1o → ∃𝑥 𝐴 = {𝑥})
28 vex 3478 . . . . 5 𝑥 ∈ V
2928ensn1 9013 . . . 4 {𝑥} ≈ 1o
30 breq1 5150 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1o ↔ {𝑥} ≈ 1o))
3129, 30mpbiri 257 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1o)
3231exlimiv 1933 . 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 396   = wceq 1541  wex 1781  wcel 2106  Vcvv 3474  c0 4321  {csn 4627  cop 4633   class class class wbr 5147  ccnv 5674  ran crn 5676  wf 6536  ontowfo 6538  1-1-ontowf1o 6539  cfv 6540  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-10 2137  ax-12 2171  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-nf 1786  df-sb 2068  df-mo 2534  df-eu 2563  df-clab 2710  df-cleq 2724  df-clel 2810  df-ne 2941  df-ral 3062  df-rex 3071  df-reu 3377  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-uni 4908  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-res 5687  df-ima 5688  df-suc 6367  df-iota 6492  df-fun 6542  df-fn 6543  df-f 6544  df-f1 6545  df-fo 6546  df-f1o 6547  df-fv 6548  df-1o 8462  df-en 8936
This theorem is referenced by:  en1b  9019  en1bOLD  9020  reuen1  9021  en1eqsn  9270  en2  9277  card1  9959  pm54.43  9992  hash1elsn  14327  hash1snb  14375  ufildom1  23421  unidifsnel  31759  unidifsnne  31760  funen1cnv  34079  lfuhgr3  34098  snen1g  42260
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