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Theorem fidcenum 7001
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 6997 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
2 simpll 527 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
3 simpr 110 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝑓:𝑛onto𝐴)
4 simplr 528 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝑛 ∈ ω)
52, 3, 4fidcenumlemr 7000 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝐴 ∈ Fin)
65ex 115 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (𝑓:𝑛onto𝐴𝐴 ∈ Fin))
76exlimdv 1830 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (∃𝑓 𝑓:𝑛onto𝐴𝐴 ∈ Fin))
87rexlimdva 2607 . . 3 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → (∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴𝐴 ∈ Fin))
98imp 124 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴) → 𝐴 ∈ Fin)
101, 9impbii 126 1 (𝐴 ∈ Fin ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
Colors of variables: wff set class
Syntax hints:  wa 104  wb 105  DECID wdc 835  wex 1503  wcel 2160  wral 2468  wrex 2469  ωcom 4614  ontowfo 5240  Fincfn 6781
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4143  ax-nul 4151  ax-pow 4199  ax-pr 4234  ax-un 4458  ax-setind 4561  ax-iinf 4612
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ne 2361  df-ral 2473  df-rex 2474  df-v 2758  df-sbc 2982  df-dif 3151  df-un 3153  df-in 3155  df-ss 3162  df-nul 3443  df-pw 3599  df-sn 3620  df-pr 3621  df-op 3623  df-uni 3832  df-int 3867  df-br 4026  df-opab 4087  df-tr 4124  df-id 4318  df-iord 4391  df-on 4393  df-suc 4396  df-iom 4615  df-xp 4657  df-rel 4658  df-cnv 4659  df-co 4660  df-dm 4661  df-rn 4662  df-res 4663  df-ima 4664  df-iota 5203  df-fun 5244  df-fn 5245  df-f 5246  df-f1 5247  df-fo 5248  df-f1o 5249  df-fv 5250  df-1o 6456  df-er 6574  df-en 6782  df-fin 6784
This theorem is referenced by:  finct  7161  ctinf  12561
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