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Theorem rp-isfinite6 42571
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 891 . . . 4 (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)
21biantrur 529 . . 3 (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3 andir 1005 . . 3 (((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin) ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
42, 3bitri 274 . 2 (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
5 simpl 481 . . . 4 ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) → 𝐴 = ∅)
6 0fin 9173 . . . . . 6 ∅ ∈ Fin
7 eleq1a 2826 . . . . . 6 (∅ ∈ Fin → (𝐴 = ∅ → 𝐴 ∈ Fin))
86, 7ax-mp 5 . . . . 5 (𝐴 = ∅ → 𝐴 ∈ Fin)
98ancli 547 . . . 4 (𝐴 = ∅ → (𝐴 = ∅ ∧ 𝐴 ∈ Fin))
105, 9impbii 208 . . 3 ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ 𝐴 = ∅)
11 rp-isfinite5 42570 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12 df-rex 3069 . . . . . 6 (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
1311, 12bitri 274 . . . . 5 (𝐴 ∈ Fin ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
1413anbi2i 621 . . . 4 ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
15 df-rex 3069 . . . . 5 (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16 en0 9015 . . . . . . . . . . . . . 14 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
17 ensymb 9000 . . . . . . . . . . . . . 14 (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
1816, 17bitr3i 276 . . . . . . . . . . . . 13 (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
1918notbii 319 . . . . . . . . . . . 12 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
20 elnn0 12478 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
2120anbi1i 622 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴))
22 andir 1005 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2321, 22bitri 274 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2419, 23anbi12i 625 . . . . . . . . . . 11 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
25 andi 1004 . . . . . . . . . . 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 274 . . . . . . . . . 10 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
27 3anass 1093 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)))
28 3anass 1093 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2927, 28orbi12i 911 . . . . . . . . . 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 237 . . . . . . . . 9 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
31 simp2 1135 . . . . . . . . . 10 ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
32 oveq2 7419 . . . . . . . . . . . 12 (𝑛 = 0 → (1...𝑛) = (1...0))
33 fz10 13526 . . . . . . . . . . . 12 (1...0) = ∅
3432, 33eqtrdi 2786 . . . . . . . . . . 11 (𝑛 = 0 → (1...𝑛) = ∅)
35 simp2 1135 . . . . . . . . . . . . 13 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) = ∅)
36 simp3 1136 . . . . . . . . . . . . 13 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) ≈ 𝐴)
3735, 36eqbrtrrd 5171 . . . . . . . . . . . 12 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ∅ ≈ 𝐴)
38 simp1 1134 . . . . . . . . . . . 12 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
3937, 38pm2.21dd 194 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
4034, 39syl3an2 1162 . . . . . . . . . 10 ((¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
4131, 40jaoi 853 . . . . . . . . 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 769 . . . . . . . 8 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (1...𝑛) ≈ 𝐴)
4442, 43jca 510 . . . . . . 7 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
45 nngt0 12247 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
46 hash0 14331 . . . . . . . . . . . . 13 (♯‘∅) = 0
4746a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (♯‘∅) = 0)
48 nnnn0 12483 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
49 hashfz1 14310 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
5048, 49syl 17 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (♯‘(1...𝑛)) = 𝑛)
5145, 47, 503brtr4d 5179 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (♯‘∅) < (♯‘(1...𝑛)))
52 fzfi 13941 . . . . . . . . . . . 12 (1...𝑛) ∈ Fin
53 hashsdom 14345 . . . . . . . . . . . 12 ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
546, 52, 53mp2an 688 . . . . . . . . . . 11 ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
5551, 54sylib 217 . . . . . . . . . 10 (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
5655anim1i 613 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
57 sdomentr 9113 . . . . . . . . . . 11 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ∅ ≺ 𝐴)
58 sdomnen 8979 . . . . . . . . . . 11 (∅ ≺ 𝐴 → ¬ ∅ ≈ 𝐴)
5957, 58syl 17 . . . . . . . . . 10 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
60 en0r 9018 . . . . . . . . . . 11 (∅ ≈ 𝐴𝐴 = ∅)
6160notbii 319 . . . . . . . . . 10 (¬ ∅ ≈ 𝐴 ↔ ¬ 𝐴 = ∅)
6259, 61sylib 217 . . . . . . . . 9 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
6356, 62syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
6448anim1i 613 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
6563, 64jca 510 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
6644, 65impbii 208 . . . . . 6 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
6766exbii 1848 . . . . 5 (∃𝑛𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
68 19.42v 1955 . . . . 5 (∃𝑛𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
6915, 67, 683bitr2ri 299 . . . 4 ((¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
7014, 69bitri 274 . . 3 ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
7110, 70orbi12i 911 . 2 (((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)) ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
724, 71bitri 274 1 (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 394  wo 843  w3a 1085   = wceq 1539  wex 1779  wcel 2104  wrex 3068  c0 4321   class class class wbr 5147  cfv 6542  (class class class)co 7411  cen 8938  csdm 8940  Fincfn 8941  0cc0 11112  1c1 11113   < clt 11252  cn 12216  0cn0 12476  ...cfz 13488  chash 14294
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-10 2135  ax-11 2152  ax-12 2169  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pow 5362  ax-pr 5426  ax-un 7727  ax-cnex 11168  ax-resscn 11169  ax-1cn 11170  ax-icn 11171  ax-addcl 11172  ax-addrcl 11173  ax-mulcl 11174  ax-mulrcl 11175  ax-mulcom 11176  ax-addass 11177  ax-mulass 11178  ax-distr 11179  ax-i2m1 11180  ax-1ne0 11181  ax-1rid 11182  ax-rnegex 11183  ax-rrecex 11184  ax-cnre 11185  ax-pre-lttri 11186  ax-pre-lttrn 11187  ax-pre-ltadd 11188  ax-pre-mulgt0 11189
This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2532  df-eu 2561  df-clab 2708  df-cleq 2722  df-clel 2808  df-nfc 2883  df-ne 2939  df-nel 3045  df-ral 3060  df-rex 3069  df-reu 3375  df-rab 3431  df-v 3474  df-sbc 3777  df-csb 3893  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-int 4950  df-iun 4998  df-br 5148  df-opab 5210  df-mpt 5231  df-tr 5265  df-id 5573  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-xp 5681  df-rel 5682  df-cnv 5683  df-co 5684  df-dm 5685  df-rn 5686  df-res 5687  df-ima 5688  df-pred 6299  df-ord 6366  df-on 6367  df-lim 6368  df-suc 6369  df-iota 6494  df-fun 6544  df-fn 6545  df-f 6546  df-f1 6547  df-fo 6548  df-f1o 6549  df-fv 6550  df-riota 7367  df-ov 7414  df-oprab 7415  df-mpo 7416  df-om 7858  df-1st 7977  df-2nd 7978  df-frecs 8268  df-wrecs 8299  df-recs 8373  df-rdg 8412  df-1o 8468  df-oadd 8472  df-er 8705  df-en 8942  df-dom 8943  df-sdom 8944  df-fin 8945  df-card 9936  df-pnf 11254  df-mnf 11255  df-xr 11256  df-ltxr 11257  df-le 11258  df-sub 11450  df-neg 11451  df-nn 12217  df-n0 12477  df-xnn0 12549  df-z 12563  df-uz 12827  df-fz 13489  df-hash 14295
This theorem is referenced by: (None)
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