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Theorem fidcenum 6837
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 6833 . 2 (𝐴 ∈ Fin → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑛 ∈ ω ∃𝑓 𝑓:𝑛onto𝐴))
2 simpll 518 . . . . . . 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 519 . . . . . . 7 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝑛 ∈ ω)
52, 3, 4fidcenumlemr 6836 . . . . . 6 (((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) ∧ 𝑓:𝑛onto𝐴) → 𝐴 ∈ Fin)
65ex 114 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (𝑓:𝑛onto𝐴𝐴 ∈ Fin))
76exlimdv 1791 . . . 4 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦𝑛 ∈ ω) → (∃𝑓 𝑓:𝑛onto𝐴𝐴 ∈ Fin))
87rexlimdva 2547 . . 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 819  wex 1468  wcel 1480  wral 2414  wrex 2415  ωcom 4499  ontowfo 5116  Fincfn 6627
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-nul 4049  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-iinf 4497
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-ral 2419  df-rex 2420  df-v 2683  df-sbc 2905  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-int 3767  df-br 3925  df-opab 3985  df-tr 4022  df-id 4210  df-iord 4283  df-on 4285  df-suc 4288  df-iom 4500  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-rn 4545  df-res 4546  df-ima 4547  df-iota 5083  df-fun 5120  df-fn 5121  df-f 5122  df-f1 5123  df-fo 5124  df-f1o 5125  df-fv 5126  df-1o 6306  df-er 6422  df-en 6628  df-fin 6630
This theorem is referenced by:  finct  6994  ctinf  11932
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