ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fidcenum GIF version

Theorem fidcenum 6719
Description: A set is finite if and only if it has decidable equality and is finitely enumerable. Proposition 8.1.11 of [AczelRathjen], p. 72. The definition of "finitely enumerable" as 𝑛 ∈ ω∃𝑓𝑓:𝑛onto𝐴 is Definition 8.1.4 of [AczelRathjen], p. 71. (Contributed by Jim Kingdon, 19-Oct-2022.)
Assertion
Ref Expression
fidcenum (𝐴 ∈ Fin ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
Distinct variable group:   𝐴,𝑓,𝑛,𝑥,𝑦

Proof of Theorem fidcenum
StepHypRef Expression
1 fidcenumlemim 6715 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
2 simpll 497 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
3 simpr 109 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝑓:𝑛onto𝐴)
4 simplr 498 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝑛 ∈ ω)
52, 3, 4fidcenumlemr 6718 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝐴 ∈ Fin)
65ex 114 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (𝑓:𝑛onto𝐴𝐴 ∈ Fin))
76exlimdv 1748 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (∃𝑓 𝑓:𝑛onto𝐴𝐴 ∈ Fin))
87rexlimdva 2490 . . 3 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴𝐴 ∈ Fin))
98imp 123 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴) → 𝐴 ∈ Fin)
101, 9impbii 125 1 (𝐴 ∈ Fin ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  DECID wdc 781  wex 1427  wcel 1439  wral 2360  wrex 2361  ωcom 4418  ontowfo 5026  Fincfn 6511
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 580  ax-in2 581  ax-io 666  ax-5 1382  ax-7 1383  ax-gen 1384  ax-ie1 1428  ax-ie2 1429  ax-8 1441  ax-10 1442  ax-11 1443  ax-i12 1444  ax-bndl 1445  ax-4 1446  ax-13 1450  ax-14 1451  ax-17 1465  ax-i9 1469  ax-ial 1473  ax-i5r 1474  ax-ext 2071  ax-sep 3963  ax-nul 3971  ax-pow 4015  ax-pr 4045  ax-un 4269  ax-setind 4366  ax-iinf 4416
This theorem depends on definitions:  df-bi 116  df-dc 782  df-3or 926  df-3an 927  df-tru 1293  df-fal 1296  df-nf 1396  df-sb 1694  df-eu 1952  df-mo 1953  df-clab 2076  df-cleq 2082  df-clel 2085  df-nfc 2218  df-ne 2257  df-ral 2365  df-rex 2366  df-v 2622  df-sbc 2842  df-dif 3002  df-un 3004  df-in 3006  df-ss 3013  df-nul 3288  df-pw 3435  df-sn 3456  df-pr 3457  df-op 3459  df-uni 3660  df-int 3695  df-br 3852  df-opab 3906  df-tr 3943  df-id 4129  df-iord 4202  df-on 4204  df-suc 4207  df-iom 4419  df-xp 4458  df-rel 4459  df-cnv 4460  df-co 4461  df-dm 4462  df-rn 4463  df-res 4464  df-ima 4465  df-iota 4993  df-fun 5030  df-fn 5031  df-f 5032  df-f1 5033  df-fo 5034  df-f1o 5035  df-fv 5036  df-1o 6195  df-er 6306  df-en 6512  df-fin 6514
This theorem is referenced by:  finct  6848
  Copyright terms: Public domain W3C validator