ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  fin0or GIF version

Theorem fin0or 6918
Description: A finite set is either empty or inhabited. (Contributed by Jim Kingdon, 30-Sep-2021.)
Assertion
Ref Expression
fin0or (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥𝐴))
Distinct variable group:   𝑥,𝐴

Proof of Theorem fin0or
Dummy variables 𝑓 𝑚 𝑛 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 isfi 6791 . . 3 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴𝑛)
21biimpi 120 . 2 (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴𝑛)
3 nn0suc 4624 . . . 4 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
43ad2antrl 490 . . 3 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
5 simplrr 536 . . . . . . 7 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴𝑛)
6 simpr 110 . . . . . . 7 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝑛 = ∅)
75, 6breqtrd 4047 . . . . . 6 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8 en0 6825 . . . . . 6 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
97, 8sylib 122 . . . . 5 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
109ex 115 . . . 4 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝑛 = ∅ → 𝐴 = ∅))
11 simplrr 536 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) → 𝐴𝑛)
1211adantr 276 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴𝑛)
1312ensymd 6813 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑛𝐴)
14 bren 6777 . . . . . . . 8 (𝑛𝐴 ↔ ∃𝑓 𝑓:𝑛1-1-onto𝐴)
1513, 14sylib 122 . . . . . . 7 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑓 𝑓:𝑛1-1-onto𝐴)
16 f1of 5483 . . . . . . . . . 10 (𝑓:𝑛1-1-onto𝐴𝑓:𝑛𝐴)
1716adantl 277 . . . . . . . . 9 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑓:𝑛𝐴)
18 sucidg 4437 . . . . . . . . . . 11 (𝑚 ∈ ω → 𝑚 ∈ suc 𝑚)
1918ad3antlr 493 . . . . . . . . . 10 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑚 ∈ suc 𝑚)
20 simplr 528 . . . . . . . . . 10 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑛 = suc 𝑚)
2119, 20eleqtrrd 2269 . . . . . . . . 9 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑚𝑛)
2217, 21ffvelcdmd 5676 . . . . . . . 8 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → (𝑓𝑚) ∈ 𝐴)
23 elex2 2768 . . . . . . . 8 ((𝑓𝑚) ∈ 𝐴 → ∃𝑥 𝑥𝐴)
2422, 23syl 14 . . . . . . 7 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → ∃𝑥 𝑥𝐴)
2515, 24exlimddv 1910 . . . . . 6 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑥 𝑥𝐴)
2625ex 115 . . . . 5 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) → (𝑛 = suc 𝑚 → ∃𝑥 𝑥𝐴))
2726rexlimdva 2607 . . . 4 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (∃𝑚 ∈ ω 𝑛 = suc 𝑚 → ∃𝑥 𝑥𝐴))
2810, 27orim12d 787 . . 3 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → ((𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚) → (𝐴 = ∅ ∨ ∃𝑥 𝑥𝐴)))
294, 28mpd 13 . 2 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝐴 = ∅ ∨ ∃𝑥 𝑥𝐴))
302, 29rexlimddv 2612 1 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥𝐴))
Colors of variables: wff set class
Syntax hints:  wi 4  wa 104  wo 709   = wceq 1364  wex 1503  wcel 2160  wrex 2469  c0 3437   class class class wbr 4021  suc csuc 4386  ωcom 4610  wf 5234  1-1-ontowf1o 5237  cfv 5238  cen 6768  Fincfn 6770
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2162  ax-14 2163  ax-ext 2171  ax-sep 4139  ax-nul 4147  ax-pow 4195  ax-pr 4230  ax-un 4454  ax-iinf 4608
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2041  df-mo 2042  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-rex 2474  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3595  df-sn 3616  df-pr 3617  df-op 3619  df-uni 3828  df-int 3863  df-br 4022  df-opab 4083  df-id 4314  df-suc 4392  df-iom 4611  df-xp 4653  df-rel 4654  df-cnv 4655  df-co 4656  df-dm 4657  df-rn 4658  df-res 4659  df-ima 4660  df-iota 5199  df-fun 5240  df-fn 5241  df-f 5242  df-f1 5243  df-fo 5244  df-f1o 5245  df-fv 5246  df-er 6563  df-en 6771  df-fin 6773
This theorem is referenced by:  xpfi  6962  fival  7003  fiubm  10849  fsumcllem  11448  fprodcllem  11655
  Copyright terms: Public domain W3C validator