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Theorem fin0or 6784
 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 6659 . . 3 (𝐴 ∈ Fin ↔ ∃𝑛 ∈ ω 𝐴𝑛)
21biimpi 119 . 2 (𝐴 ∈ Fin → ∃𝑛 ∈ ω 𝐴𝑛)
3 nn0suc 4522 . . . 4 (𝑛 ∈ ω → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
43ad2antrl 482 . . 3 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝑛 = ∅ ∨ ∃𝑚 ∈ ω 𝑛 = suc 𝑚))
5 simplrr 526 . . . . . . 7 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴𝑛)
6 simpr 109 . . . . . . 7 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝑛 = ∅)
75, 6breqtrd 3958 . . . . . 6 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 ≈ ∅)
8 en0 6693 . . . . . 6 (𝐴 ≈ ∅ ↔ 𝐴 = ∅)
97, 8sylib 121 . . . . 5 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑛 = ∅) → 𝐴 = ∅)
109ex 114 . . . 4 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (𝑛 = ∅ → 𝐴 = ∅))
11 simplrr 526 . . . . . . . . . 10 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) → 𝐴𝑛)
1211adantr 274 . . . . . . . . 9 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝐴𝑛)
1312ensymd 6681 . . . . . . . 8 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → 𝑛𝐴)
14 bren 6645 . . . . . . . 8 (𝑛𝐴 ↔ ∃𝑓 𝑓:𝑛1-1-onto𝐴)
1513, 14sylib 121 . . . . . . 7 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑓 𝑓:𝑛1-1-onto𝐴)
16 f1of 5371 . . . . . . . . . 10 (𝑓:𝑛1-1-onto𝐴𝑓:𝑛𝐴)
1716adantl 275 . . . . . . . . 9 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑓:𝑛𝐴)
18 sucidg 4342 . . . . . . . . . . 11 (𝑚 ∈ ω → 𝑚 ∈ suc 𝑚)
1918ad3antlr 485 . . . . . . . . . 10 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑚 ∈ suc 𝑚)
20 simplr 520 . . . . . . . . . 10 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑛 = suc 𝑚)
2119, 20eleqtrrd 2220 . . . . . . . . 9 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → 𝑚𝑛)
2217, 21ffvelrnd 5560 . . . . . . . 8 (((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) ∧ 𝑓:𝑛1-1-onto𝐴) → (𝑓𝑚) ∈ 𝐴)
23 elex2 2703 . . . . . . . 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 1871 . . . . . 6 ((((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) ∧ 𝑛 = suc 𝑚) → ∃𝑥 𝑥𝐴)
2625ex 114 . . . . 5 (((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) ∧ 𝑚 ∈ ω) → (𝑛 = suc 𝑚 → ∃𝑥 𝑥𝐴))
2726rexlimdva 2550 . . . 4 ((𝐴 ∈ Fin ∧ (𝑛 ∈ ω ∧ 𝐴𝑛)) → (∃𝑚 ∈ ω 𝑛 = suc 𝑚 → ∃𝑥 𝑥𝐴))
2810, 27orim12d 776 . . 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 2555 1 (𝐴 ∈ Fin → (𝐴 = ∅ ∨ ∃𝑥 𝑥𝐴))
 Colors of variables: wff set class Syntax hints:   → wi 4   ∧ wa 103   ∨ wo 698   = wceq 1332  ∃wex 1469   ∈ wcel 1481  ∃wrex 2418  ∅c0 3364   class class class wbr 3933  suc csuc 4291  ωcom 4508  ⟶wf 5123  –1-1-onto→wf1o 5126  ‘cfv 5127   ≈ cen 6636  Fincfn 6638 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 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4050  ax-nul 4058  ax-pow 4102  ax-pr 4135  ax-un 4359  ax-iinf 4506 This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2689  df-sbc 2911  df-dif 3074  df-un 3076  df-in 3078  df-ss 3085  df-nul 3365  df-pw 3513  df-sn 3534  df-pr 3535  df-op 3537  df-uni 3741  df-int 3776  df-br 3934  df-opab 3994  df-id 4219  df-suc 4297  df-iom 4509  df-xp 4549  df-rel 4550  df-cnv 4551  df-co 4552  df-dm 4553  df-rn 4554  df-res 4555  df-ima 4556  df-iota 5092  df-fun 5129  df-fn 5130  df-f 5131  df-f1 5132  df-fo 5133  df-f1o 5134  df-fv 5135  df-er 6433  df-en 6639  df-fin 6641 This theorem is referenced by:  xpfi  6822  fival  6862  fsumcllem  11196
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