Users' Mathboxes Mathbox for Richard Penner < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  rp-isfinite6 Structured version   Visualization version   GIF version

Theorem rp-isfinite6 40058
Description: A set is said to be finite if it is either empty or it can be put in one-to-one correspondence with all the natural numbers between 1 and some 𝑛 ∈ ℕ. (Contributed by RP, 10-Mar-2020.)
Assertion
Ref Expression
rp-isfinite6 (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
Distinct variable group:   𝐴,𝑛

Proof of Theorem rp-isfinite6
StepHypRef Expression
1 exmid 892 . . . 4 (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)
21biantrur 534 . . 3 (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3 andir 1006 . . 3 (((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin) ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
42, 3bitri 278 . 2 (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
5 simpl 486 . . . 4 ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) → 𝐴 = ∅)
6 0fin 8730 . . . . . 6 ∅ ∈ Fin
7 eleq1a 2911 . . . . . 6 (∅ ∈ Fin → (𝐴 = ∅ → 𝐴 ∈ Fin))
86, 7ax-mp 5 . . . . 5 (𝐴 = ∅ → 𝐴 ∈ Fin)
98ancli 552 . . . 4 (𝐴 = ∅ → (𝐴 = ∅ ∧ 𝐴 ∈ Fin))
105, 9impbii 212 . . 3 ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ 𝐴 = ∅)
11 rp-isfinite5 40057 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12 df-rex 3138 . . . . . 6 (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
1311, 12bitri 278 . . . . 5 (𝐴 ∈ Fin ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
1413anbi2i 625 . . . 4 ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
15 df-rex 3138 . . . . 5 (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16 en0 8555 . . . . . . . . . . . . . 14 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
17 ensymb 8540 . . . . . . . . . . . . . 14 (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
1816, 17bitr3i 280 . . . . . . . . . . . . 13 (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
1918notbii 323 . . . . . . . . . . . 12 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
20 elnn0 11885 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
2120anbi1i 626 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴))
22 andir 1006 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2321, 22bitri 278 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2419, 23anbi12i 629 . . . . . . . . . . 11 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
25 andi 1005 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
2624, 25bitri 278 . . . . . . . . . 10 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
27 3anass 1092 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)))
28 3anass 1092 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2927, 28orbi12i 912 . . . . . . . . . 10 (((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
3026, 29sylbb2 241 . . . . . . . . 9 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
31 simp2 1134 . . . . . . . . . 10 ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
32 oveq2 7146 . . . . . . . . . . . 12 (𝑛 = 0 → (1...𝑛) = (1...0))
33 fz10 12921 . . . . . . . . . . . 12 (1...0) = ∅
3432, 33syl6eq 2875 . . . . . . . . . . 11 (𝑛 = 0 → (1...𝑛) = ∅)
35 simp2 1134 . . . . . . . . . . . . 13 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) = ∅)
36 simp3 1135 . . . . . . . . . . . . 13 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) ≈ 𝐴)
3735, 36eqbrtrrd 5071 . . . . . . . . . . . 12 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ∅ ≈ 𝐴)
38 simp1 1133 . . . . . . . . . . . 12 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
3937, 38pm2.21dd 198 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
4034, 39syl3an2 1161 . . . . . . . . . 10 ((¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
4131, 40jaoi 854 . . . . . . . . 9 (((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
4230, 41syl 17 . . . . . . . 8 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
43 simprr 772 . . . . . . . 8 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (1...𝑛) ≈ 𝐴)
4442, 43jca 515 . . . . . . 7 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
45 nngt0 11654 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
46 hash0 13722 . . . . . . . . . . . . 13 (♯‘∅) = 0
4746a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (♯‘∅) = 0)
48 nnnn0 11890 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
49 hashfz1 13700 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
5048, 49syl 17 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (♯‘(1...𝑛)) = 𝑛)
5145, 47, 503brtr4d 5079 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (♯‘∅) < (♯‘(1...𝑛)))
52 fzfi 13333 . . . . . . . . . . . 12 (1...𝑛) ∈ Fin
53 hashsdom 13736 . . . . . . . . . . . 12 ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
546, 52, 53mp2an 691 . . . . . . . . . . 11 ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
5551, 54sylib 221 . . . . . . . . . 10 (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
5655anim1i 617 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
57 sdomentr 8635 . . . . . . . . . . 11 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ∅ ≺ 𝐴)
58 sdomnen 8521 . . . . . . . . . . 11 (∅ ≺ 𝐴 → ¬ ∅ ≈ 𝐴)
5957, 58syl 17 . . . . . . . . . 10 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
60 ensymb 8540 . . . . . . . . . . . 12 (∅ ≈ 𝐴𝐴 ≈ ∅)
6160, 16bitri 278 . . . . . . . . . . 11 (∅ ≈ 𝐴𝐴 = ∅)
6261notbii 323 . . . . . . . . . 10 (¬ ∅ ≈ 𝐴 ↔ ¬ 𝐴 = ∅)
6359, 62sylib 221 . . . . . . . . 9 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
6456, 63syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
6548anim1i 617 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
6664, 65jca 515 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
6744, 66impbii 212 . . . . . 6 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
6867exbii 1849 . . . . 5 (∃𝑛𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
69 19.42v 1955 . . . . 5 (∃𝑛𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
7015, 68, 693bitr2ri 303 . . . 4 ((¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
7114, 70bitri 278 . . 3 ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
7210, 71orbi12i 912 . 2 (((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)) ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
734, 72bitri 278 1 (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wa 399  wo 844  w3a 1084   = wceq 1538  wex 1781  wcel 2115  wrex 3133  c0 4274   class class class wbr 5047  cfv 6336  (class class class)co 7138  cen 8489  csdm 8491  Fincfn 8492  0cc0 10522  1c1 10523   < clt 10660  cn 11623  0cn0 11883  ...cfz 12883  chash 13684
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2179  ax-ext 2796  ax-sep 5184  ax-nul 5191  ax-pow 5247  ax-pr 5311  ax-un 7444  ax-cnex 10578  ax-resscn 10579  ax-1cn 10580  ax-icn 10581  ax-addcl 10582  ax-addrcl 10583  ax-mulcl 10584  ax-mulrcl 10585  ax-mulcom 10586  ax-addass 10587  ax-mulass 10588  ax-distr 10589  ax-i2m1 10590  ax-1ne0 10591  ax-1rid 10592  ax-rnegex 10593  ax-rrecex 10594  ax-cnre 10595  ax-pre-lttri 10596  ax-pre-lttrn 10597  ax-pre-ltadd 10598  ax-pre-mulgt0 10599
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-mo 2624  df-eu 2655  df-clab 2803  df-cleq 2817  df-clel 2896  df-nfc 2964  df-ne 3014  df-nel 3118  df-ral 3137  df-rex 3138  df-reu 3139  df-rab 3141  df-v 3481  df-sbc 3758  df-csb 3866  df-dif 3921  df-un 3923  df-in 3925  df-ss 3935  df-pss 3937  df-nul 4275  df-if 4449  df-pw 4522  df-sn 4549  df-pr 4551  df-tp 4553  df-op 4555  df-uni 4820  df-int 4858  df-iun 4902  df-br 5048  df-opab 5110  df-mpt 5128  df-tr 5154  df-id 5441  df-eprel 5446  df-po 5455  df-so 5456  df-fr 5495  df-we 5497  df-xp 5542  df-rel 5543  df-cnv 5544  df-co 5545  df-dm 5546  df-rn 5547  df-res 5548  df-ima 5549  df-pred 6129  df-ord 6175  df-on 6176  df-lim 6177  df-suc 6178  df-iota 6295  df-fun 6338  df-fn 6339  df-f 6340  df-f1 6341  df-fo 6342  df-f1o 6343  df-fv 6344  df-riota 7096  df-ov 7141  df-oprab 7142  df-mpo 7143  df-om 7564  df-1st 7672  df-2nd 7673  df-wrecs 7930  df-recs 7991  df-rdg 8029  df-1o 8085  df-oadd 8089  df-er 8272  df-en 8493  df-dom 8494  df-sdom 8495  df-fin 8496  df-card 9352  df-pnf 10662  df-mnf 10663  df-xr 10664  df-ltxr 10665  df-le 10666  df-sub 10857  df-neg 10858  df-nn 11624  df-n0 11884  df-xnn0 11954  df-z 11968  df-uz 12230  df-fz 12884  df-hash 13685
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator