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Theorem fidcenum 6900
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 6896 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
2 simpll 519 . . . . . . 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 520 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝑛 ∈ ω)
52, 3, 4fidcenumlemr 6899 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝐴 ∈ Fin)
65ex 114 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (𝑓:𝑛onto𝐴𝐴 ∈ Fin))
76exlimdv 1799 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (∃𝑓 𝑓:𝑛onto𝐴𝐴 ∈ Fin))
87rexlimdva 2574 . . 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 820  wex 1472  wcel 2128  wral 2435  wrex 2436  ωcom 4549  ontowfo 5168  Fincfn 6685
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1427  ax-7 1428  ax-gen 1429  ax-ie1 1473  ax-ie2 1474  ax-8 1484  ax-10 1485  ax-11 1486  ax-i12 1487  ax-bndl 1489  ax-4 1490  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-13 2130  ax-14 2131  ax-ext 2139  ax-sep 4082  ax-nul 4090  ax-pow 4135  ax-pr 4169  ax-un 4393  ax-setind 4496  ax-iinf 4547
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  df-3an 965  df-tru 1338  df-fal 1341  df-nf 1441  df-sb 1743  df-eu 2009  df-mo 2010  df-clab 2144  df-cleq 2150  df-clel 2153  df-nfc 2288  df-ne 2328  df-ral 2440  df-rex 2441  df-v 2714  df-sbc 2938  df-dif 3104  df-un 3106  df-in 3108  df-ss 3115  df-nul 3395  df-pw 3545  df-sn 3566  df-pr 3567  df-op 3569  df-uni 3773  df-int 3808  df-br 3966  df-opab 4026  df-tr 4063  df-id 4253  df-iord 4326  df-on 4328  df-suc 4331  df-iom 4550  df-xp 4592  df-rel 4593  df-cnv 4594  df-co 4595  df-dm 4596  df-rn 4597  df-res 4598  df-ima 4599  df-iota 5135  df-fun 5172  df-fn 5173  df-f 5174  df-f1 5175  df-fo 5176  df-f1o 5177  df-fv 5178  df-1o 6363  df-er 6480  df-en 6686  df-fin 6688
This theorem is referenced by:  finct  7060  ctinf  12170
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