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Theorem ennnfone 11844
Description: A condition for a set being countably infinite. Corollary 8.1.13 of [AczelRathjen], p. 73. Roughly speaking, the condition says that 𝐴 is countable (that's the 𝑓:ℕ0onto𝐴 part, as seen in theorems like ctm 6960), infinite (that's the part about being able to find an element of 𝐴 distinct from any mapping of a natural number via 𝑓), and has decidable equality. (Contributed by Jim Kingdon, 27-Oct-2022.)
Assertion
Ref Expression
ennnfone (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))))
Distinct variable groups:   𝐴,𝑓,𝑗,𝑛,𝑥,𝑦   𝑓,𝑘,𝑗,𝑛
Allowed substitution hint:   𝐴(𝑘)

Proof of Theorem ennnfone
StepHypRef Expression
1 ennnfonelemim 11843 . 2 (𝐴 ≈ ℕ → (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))))
2 simpl 108 . . . . . 6 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))) → ∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦)
3 simprl 503 . . . . . 6 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))) → 𝑓:ℕ0onto𝐴)
4 simprr 504 . . . . . 6 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))) → ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))
52, 3, 4ennnfonelemr 11842 . . . . 5 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ (𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))) → 𝐴 ≈ ℕ)
65ex 114 . . . 4 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → ((𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗)) → 𝐴 ≈ ℕ))
76exlimdv 1773 . . 3 (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 → (∃𝑓(𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗)) → 𝐴 ≈ ℕ))
87imp 123 . 2 ((∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))) → 𝐴 ≈ ℕ)
91, 8impbii 125 1 (𝐴 ≈ ℕ ↔ (∀𝑥𝐴𝑦𝐴 DECID 𝑥 = 𝑦 ∧ ∃𝑓(𝑓:ℕ0onto𝐴 ∧ ∀𝑛 ∈ ℕ0𝑘 ∈ ℕ0𝑗 ∈ (0...𝑛)(𝑓𝑘) ≠ (𝑓𝑗))))
Colors of variables: wff set class
Syntax hints:  wa 103  wb 104  DECID wdc 802  wex 1451  wne 2283  wral 2391  wrex 2392   class class class wbr 3897  ontowfo 5089  cfv 5091  (class class class)co 5740  cen 6598  0cc0 7584  cn 8680  0cn0 8931  ...cfz 9741
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 586  ax-in2 587  ax-io 681  ax-5 1406  ax-7 1407  ax-gen 1408  ax-ie1 1452  ax-ie2 1453  ax-8 1465  ax-10 1466  ax-11 1467  ax-i12 1468  ax-bndl 1469  ax-4 1470  ax-13 1474  ax-14 1475  ax-17 1489  ax-i9 1493  ax-ial 1497  ax-i5r 1498  ax-ext 2097  ax-coll 4011  ax-sep 4014  ax-nul 4022  ax-pow 4066  ax-pr 4099  ax-un 4323  ax-setind 4420  ax-iinf 4470  ax-cnex 7675  ax-resscn 7676  ax-1cn 7677  ax-1re 7678  ax-icn 7679  ax-addcl 7680  ax-addrcl 7681  ax-mulcl 7682  ax-addcom 7684  ax-addass 7686  ax-distr 7688  ax-i2m1 7689  ax-0lt1 7690  ax-0id 7692  ax-rnegex 7693  ax-cnre 7695  ax-pre-ltirr 7696  ax-pre-ltwlin 7697  ax-pre-lttrn 7698  ax-pre-ltadd 7700
This theorem depends on definitions:  df-bi 116  df-dc 803  df-3or 946  df-3an 947  df-tru 1317  df-fal 1320  df-nf 1420  df-sb 1719  df-eu 1978  df-mo 1979  df-clab 2102  df-cleq 2108  df-clel 2111  df-nfc 2245  df-ne 2284  df-nel 2379  df-ral 2396  df-rex 2397  df-reu 2398  df-rab 2400  df-v 2660  df-sbc 2881  df-csb 2974  df-dif 3041  df-un 3043  df-in 3045  df-ss 3052  df-nul 3332  df-if 3443  df-pw 3480  df-sn 3501  df-pr 3502  df-op 3504  df-uni 3705  df-int 3740  df-iun 3783  df-br 3898  df-opab 3958  df-mpt 3959  df-tr 3995  df-id 4183  df-iord 4256  df-on 4258  df-ilim 4259  df-suc 4261  df-iom 4473  df-xp 4513  df-rel 4514  df-cnv 4515  df-co 4516  df-dm 4517  df-rn 4518  df-res 4519  df-ima 4520  df-iota 5056  df-fun 5093  df-fn 5094  df-f 5095  df-f1 5096  df-fo 5097  df-f1o 5098  df-fv 5099  df-riota 5696  df-ov 5743  df-oprab 5744  df-mpo 5745  df-1st 6004  df-2nd 6005  df-recs 6168  df-frec 6254  df-er 6395  df-pm 6511  df-en 6601  df-pnf 7766  df-mnf 7767  df-xr 7768  df-ltxr 7769  df-le 7770  df-sub 7899  df-neg 7900  df-inn 8681  df-n0 8932  df-z 9009  df-uz 9279  df-fz 9742  df-seqfrec 10170
This theorem is referenced by:  ctinfom  11847
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