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Theorem en1 6367
Description: A set is equinumerous to ordinal one iff it is a singleton. (Contributed by NM, 25-Jul-2004.)
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 6097 . . . . 5 1𝑜 = {∅}
21breq2i 3813 . . . 4 (𝐴 ≈ 1𝑜𝐴 ≈ {∅})
3 bren 6315 . . . 4 (𝐴 ≈ {∅} ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
42, 3bitri 182 . . 3 (𝐴 ≈ 1𝑜 ↔ ∃𝑓 𝑓:𝐴1-1-onto→{∅})
5 f1ocnv 5190 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝑓:{∅}–1-1-onto𝐴)
6 f1ofo 5184 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}–onto𝐴)
7 forn 5160 . . . . . . . 8 (𝑓:{∅}–onto𝐴 → ran 𝑓 = 𝐴)
86, 7syl 14 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = 𝐴)
9 f1of 5177 . . . . . . . . . 10 (𝑓:{∅}–1-1-onto𝐴𝑓:{∅}⟶𝐴)
10 0ex 3925 . . . . . . . . . . . 12 ∅ ∈ V
1110fsn2 5389 . . . . . . . . . . 11 (𝑓:{∅}⟶𝐴 ↔ ((𝑓‘∅) ∈ 𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩}))
1211simprbi 269 . . . . . . . . . 10 (𝑓:{∅}⟶𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
139, 12syl 14 . . . . . . . . 9 (𝑓:{∅}–1-1-onto𝐴𝑓 = {⟨∅, (𝑓‘∅)⟩})
1413rneqd 4611 . . . . . . . 8 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = ran {⟨∅, (𝑓‘∅)⟩})
1510rnsnop 4851 . . . . . . . 8 ran {⟨∅, (𝑓‘∅)⟩} = {(𝑓‘∅)}
1614, 15syl6eq 2131 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴 → ran 𝑓 = {(𝑓‘∅)})
178, 16eqtr3d 2117 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴𝐴 = {(𝑓‘∅)})
185, 17syl 14 . . . . 5 (𝑓:𝐴1-1-onto→{∅} → 𝐴 = {(𝑓‘∅)})
19 f1ofn 5178 . . . . . . 7 (𝑓:{∅}–1-1-onto𝐴𝑓 Fn {∅})
2010snid 3443 . . . . . . 7 ∅ ∈ {∅}
21 funfvex 5243 . . . . . . . 8 ((Fun 𝑓 ∧ ∅ ∈ dom 𝑓) → (𝑓‘∅) ∈ V)
2221funfni 5050 . . . . . . 7 ((𝑓 Fn {∅} ∧ ∅ ∈ {∅}) → (𝑓‘∅) ∈ V)
2319, 20, 22sylancl 404 . . . . . 6 (𝑓:{∅}–1-1-onto𝐴 → (𝑓‘∅) ∈ V)
24 sneq 3427 . . . . . . . 8 (𝑥 = (𝑓‘∅) → {𝑥} = {(𝑓‘∅)})
2524eqeq2d 2094 . . . . . . 7 (𝑥 = (𝑓‘∅) → (𝐴 = {𝑥} ↔ 𝐴 = {(𝑓‘∅)}))
2625spcegv 2695 . . . . . 6 ((𝑓‘∅) ∈ V → (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
2723, 26syl 14 . . . . 5 (𝑓:{∅}–1-1-onto𝐴 → (𝐴 = {(𝑓‘∅)} → ∃𝑥 𝐴 = {𝑥}))
285, 18, 27sylc 61 . . . 4 (𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
2928exlimiv 1530 . . 3 (∃𝑓 𝑓:𝐴1-1-onto→{∅} → ∃𝑥 𝐴 = {𝑥})
304, 29sylbi 119 . 2 (𝐴 ≈ 1𝑜 → ∃𝑥 𝐴 = {𝑥})
31 vex 2613 . . . . 5 𝑥 ∈ V
3231ensn1 6364 . . . 4 {𝑥} ≈ 1𝑜
33 breq1 3808 . . . 4 (𝐴 = {𝑥} → (𝐴 ≈ 1𝑜 ↔ {𝑥} ≈ 1𝑜))
3432, 33mpbiri 166 . . 3 (𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
3534exlimiv 1530 . 2 (∃𝑥 𝐴 = {𝑥} → 𝐴 ≈ 1𝑜)
3630, 35impbii 124 1 (𝐴 ≈ 1𝑜 ↔ ∃𝑥 𝐴 = {𝑥})
Colors of variables: wff set class
Syntax hints:  wi 4  wb 103   = wceq 1285  wex 1422  wcel 1434  Vcvv 2610  c0 3267  {csn 3416  cop 3419   class class class wbr 3805  ccnv 4390  ran crn 4392   Fn wfn 4947  wf 4948  ontowfo 4950  1-1-ontowf1o 4951  cfv 4952  1𝑜c1o 6078  cen 6306
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065  ax-sep 3916  ax-nul 3924  ax-pow 3968  ax-pr 3992  ax-un 4216
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-eu 1946  df-mo 1947  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-rex 2359  df-reu 2360  df-v 2612  df-sbc 2825  df-dif 2984  df-un 2986  df-in 2988  df-ss 2995  df-nul 3268  df-pw 3402  df-sn 3422  df-pr 3423  df-op 3425  df-uni 3622  df-br 3806  df-opab 3860  df-id 4076  df-suc 4154  df-xp 4397  df-rel 4398  df-cnv 4399  df-co 4400  df-dm 4401  df-rn 4402  df-res 4403  df-ima 4404  df-iota 4917  df-fun 4954  df-fn 4955  df-f 4956  df-f1 4957  df-fo 4958  df-f1o 4959  df-fv 4960  df-1o 6085  df-en 6309
This theorem is referenced by:  en1bg  6368  reuen1  6369  pm54.43  6570
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