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Theorem rp-isfinite6 38384
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 Richard Penner, 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 430 . . . 4 (𝐴 = ∅ ∨ ¬ 𝐴 = ∅)
21biantrur 528 . . 3 (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin))
3 andir 948 . . 3 (((𝐴 = ∅ ∨ ¬ 𝐴 = ∅) ∧ 𝐴 ∈ Fin) ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
42, 3bitri 264 . 2 (𝐴 ∈ Fin ↔ ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)))
5 simpl 474 . . . 4 ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) → 𝐴 = ∅)
6 0fin 8355 . . . . . 6 ∅ ∈ Fin
7 eleq1a 2834 . . . . . 6 (∅ ∈ Fin → (𝐴 = ∅ → 𝐴 ∈ Fin))
86, 7ax-mp 5 . . . . 5 (𝐴 = ∅ → 𝐴 ∈ Fin)
98ancli 575 . . . 4 (𝐴 = ∅ → (𝐴 = ∅ ∧ 𝐴 ∈ Fin))
105, 9impbii 199 . . 3 ((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ 𝐴 = ∅)
11 rp-isfinite5 38383 . . . . . 6 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴)
12 df-rex 3056 . . . . . 6 (∃𝑛 ∈ ℕ0 (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
1311, 12bitri 264 . . . . 5 (𝐴 ∈ Fin ↔ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
1413anbi2i 732 . . . 4 ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
15 df-rex 3056 . . . . 5 (∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴 ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
16 en0 8186 . . . . . . . . . . . . . . 15 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
1716bicomi 214 . . . . . . . . . . . . . 14 (𝐴 = ∅ ↔ 𝐴 ≈ ∅)
18 ensymb 8171 . . . . . . . . . . . . . 14 (𝐴 ≈ ∅ ↔ ∅ ≈ 𝐴)
1917, 18bitri 264 . . . . . . . . . . . . 13 (𝐴 = ∅ ↔ ∅ ≈ 𝐴)
2019notbii 309 . . . . . . . . . . . 12 𝐴 = ∅ ↔ ¬ ∅ ≈ 𝐴)
21 elnn0 11506 . . . . . . . . . . . . . 14 (𝑛 ∈ ℕ0 ↔ (𝑛 ∈ ℕ ∨ 𝑛 = 0))
2221anbi1i 733 . . . . . . . . . . . . 13 ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴))
23 andir 948 . . . . . . . . . . . . 13 (((𝑛 ∈ ℕ ∨ 𝑛 = 0) ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2422, 23bitri 264 . . . . . . . . . . . 12 ((𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴) ↔ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
2520, 24anbi12i 735 . . . . . . . . . . 11 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
26 andi 947 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴 ∧ ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
2725, 26bitri 264 . . . . . . . . . 10 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
28 3anass 1081 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)))
29 3anass 1081 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) ↔ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
3028, 29orbi12i 544 . . . . . . . . . 10 (((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ((¬ ∅ ≈ 𝐴 ∧ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴)) ∨ (¬ ∅ ≈ 𝐴 ∧ (𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴))))
3127, 30sylbb2 228 . . . . . . . . 9 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)))
32 simp2 1132 . . . . . . . . . 10 ((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
33 oveq2 6822 . . . . . . . . . . . 12 (𝑛 = 0 → (1...𝑛) = (1...0))
34 fz10 12575 . . . . . . . . . . . 12 (1...0) = ∅
3533, 34syl6eq 2810 . . . . . . . . . . 11 (𝑛 = 0 → (1...𝑛) = ∅)
36 simp2 1132 . . . . . . . . . . . . 13 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) = ∅)
37 simp3 1133 . . . . . . . . . . . . 13 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → (1...𝑛) ≈ 𝐴)
3836, 37eqbrtrrd 4828 . . . . . . . . . . . 12 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ∅ ≈ 𝐴)
39 simp1 1131 . . . . . . . . . . . 12 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
4038, 39pm2.21dd 186 . . . . . . . . . . 11 ((¬ ∅ ≈ 𝐴 ∧ (1...𝑛) = ∅ ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
4135, 40syl3an2 1168 . . . . . . . . . 10 ((¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴) → 𝑛 ∈ ℕ)
4232, 41jaoi 393 . . . . . . . . 9 (((¬ ∅ ≈ 𝐴𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) ∨ (¬ ∅ ≈ 𝐴𝑛 = 0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
4331, 42syl 17 . . . . . . . 8 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → 𝑛 ∈ ℕ)
44 simprr 813 . . . . . . . 8 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (1...𝑛) ≈ 𝐴)
4543, 44jca 555 . . . . . . 7 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) → (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
46 nngt0 11261 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → 0 < 𝑛)
47 hash0 13370 . . . . . . . . . . . . 13 (♯‘∅) = 0
4847a1i 11 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (♯‘∅) = 0)
49 nnnn0 11511 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ → 𝑛 ∈ ℕ0)
50 hashfz1 13348 . . . . . . . . . . . . 13 (𝑛 ∈ ℕ0 → (♯‘(1...𝑛)) = 𝑛)
5149, 50syl 17 . . . . . . . . . . . 12 (𝑛 ∈ ℕ → (♯‘(1...𝑛)) = 𝑛)
5246, 48, 513brtr4d 4836 . . . . . . . . . . 11 (𝑛 ∈ ℕ → (♯‘∅) < (♯‘(1...𝑛)))
53 fzfi 12985 . . . . . . . . . . . 12 (1...𝑛) ∈ Fin
54 hashsdom 13382 . . . . . . . . . . . 12 ((∅ ∈ Fin ∧ (1...𝑛) ∈ Fin) → ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛)))
556, 53, 54mp2an 710 . . . . . . . . . . 11 ((♯‘∅) < (♯‘(1...𝑛)) ↔ ∅ ≺ (1...𝑛))
5652, 55sylib 208 . . . . . . . . . 10 (𝑛 ∈ ℕ → ∅ ≺ (1...𝑛))
5756anim1i 593 . . . . . . . . 9 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴))
58 sdomentr 8261 . . . . . . . . . . 11 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ∅ ≺ 𝐴)
59 sdomnen 8152 . . . . . . . . . . 11 (∅ ≺ 𝐴 → ¬ ∅ ≈ 𝐴)
6058, 59syl 17 . . . . . . . . . 10 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ ∅ ≈ 𝐴)
61 ensymb 8171 . . . . . . . . . . . 12 (∅ ≈ 𝐴𝐴 ≈ ∅)
6261, 16bitri 264 . . . . . . . . . . 11 (∅ ≈ 𝐴𝐴 = ∅)
6362notbii 309 . . . . . . . . . 10 (¬ ∅ ≈ 𝐴 ↔ ¬ 𝐴 = ∅)
6460, 63sylib 208 . . . . . . . . 9 ((∅ ≺ (1...𝑛) ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
6557, 64syl 17 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → ¬ 𝐴 = ∅)
6649anim1i 593 . . . . . . . 8 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴))
6765, 66jca 555 . . . . . . 7 ((𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴) → (¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
6845, 67impbii 199 . . . . . 6 ((¬ 𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
6968exbii 1923 . . . . 5 (∃𝑛𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛(𝑛 ∈ ℕ ∧ (1...𝑛) ≈ 𝐴))
70 19.42v 2030 . . . . 5 (∃𝑛𝐴 = ∅ ∧ (𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ (¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)))
7115, 69, 703bitr2ri 289 . . . 4 ((¬ 𝐴 = ∅ ∧ ∃𝑛(𝑛 ∈ ℕ0 ∧ (1...𝑛) ≈ 𝐴)) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
7214, 71bitri 264 . . 3 ((¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin) ↔ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴)
7310, 72orbi12i 544 . 2 (((𝐴 = ∅ ∧ 𝐴 ∈ Fin) ∨ (¬ 𝐴 = ∅ ∧ 𝐴 ∈ Fin)) ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
744, 73bitri 264 1 (𝐴 ∈ Fin ↔ (𝐴 = ∅ ∨ ∃𝑛 ∈ ℕ (1...𝑛) ≈ 𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wo 382  wa 383  w3a 1072   = wceq 1632  wex 1853  wcel 2139  wrex 3051  c0 4058   class class class wbr 4804  cfv 6049  (class class class)co 6814  cen 8120  csdm 8122  Fincfn 8123  0cc0 10148  1c1 10149   < clt 10286  cn 11232  0cn0 11504  ...cfz 12539  chash 13331
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-8 2141  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pow 4992  ax-pr 5055  ax-un 7115  ax-cnex 10204  ax-resscn 10205  ax-1cn 10206  ax-icn 10207  ax-addcl 10208  ax-addrcl 10209  ax-mulcl 10210  ax-mulrcl 10211  ax-mulcom 10212  ax-addass 10213  ax-mulass 10214  ax-distr 10215  ax-i2m1 10216  ax-1ne0 10217  ax-1rid 10218  ax-rnegex 10219  ax-rrecex 10220  ax-cnre 10221  ax-pre-lttri 10222  ax-pre-lttrn 10223  ax-pre-ltadd 10224  ax-pre-mulgt0 10225
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3or 1073  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-eu 2611  df-mo 2612  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ne 2933  df-nel 3036  df-ral 3055  df-rex 3056  df-reu 3057  df-rab 3059  df-v 3342  df-sbc 3577  df-csb 3675  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-pss 3731  df-nul 4059  df-if 4231  df-pw 4304  df-sn 4322  df-pr 4324  df-tp 4326  df-op 4328  df-uni 4589  df-int 4628  df-iun 4674  df-br 4805  df-opab 4865  df-mpt 4882  df-tr 4905  df-id 5174  df-eprel 5179  df-po 5187  df-so 5188  df-fr 5225  df-we 5227  df-xp 5272  df-rel 5273  df-cnv 5274  df-co 5275  df-dm 5276  df-rn 5277  df-res 5278  df-ima 5279  df-pred 5841  df-ord 5887  df-on 5888  df-lim 5889  df-suc 5890  df-iota 6012  df-fun 6051  df-fn 6052  df-f 6053  df-f1 6054  df-fo 6055  df-f1o 6056  df-fv 6057  df-riota 6775  df-ov 6817  df-oprab 6818  df-mpt2 6819  df-om 7232  df-1st 7334  df-2nd 7335  df-wrecs 7577  df-recs 7638  df-rdg 7676  df-1o 7730  df-oadd 7734  df-er 7913  df-en 8124  df-dom 8125  df-sdom 8126  df-fin 8127  df-card 8975  df-pnf 10288  df-mnf 10289  df-xr 10290  df-ltxr 10291  df-le 10292  df-sub 10480  df-neg 10481  df-nn 11233  df-n0 11505  df-xnn0 11576  df-z 11590  df-uz 11900  df-fz 12540  df-hash 13332
This theorem is referenced by: (None)
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