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Theorem exmidonfin 7066
Description: If a finite ordinal is a natural number, excluded middle follows. That excluded middle implies that a finite ordinal is a natural number is proved in the Metamath Proof Explorer. That a natural number is a finite ordinal is shown at nnfi 6773 and nnon 4530. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
Assertion
Ref Expression
exmidonfin (ω = (On ∩ Fin) → EXMID)

Proof of Theorem exmidonfin
Dummy variables 𝑥 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2140 . . . 4 {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}} = {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}}
21exmidonfinlem 7065 . . 3 (ω = (On ∩ Fin) → DECID 𝑧 = {∅})
32adantr 274 . 2 ((ω = (On ∩ Fin) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅})
43exmid1dc 4130 1 (ω = (On ∩ Fin) → EXMID)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  DECID wdc 820   = wceq 1332  {crab 2421  cin 3074  wss 3075  c0 3367  {csn 3531  {cpr 3532  EXMIDwem 4125  Oncon0 4292  ωcom 4511  Fincfn 6641
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 4053  ax-nul 4061  ax-pow 4105  ax-pr 4138  ax-un 4362  ax-setind 4459  ax-iinf 4509
This theorem depends on definitions:  df-bi 116  df-dc 821  df-3or 964  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-ne 2310  df-ral 2422  df-rex 2423  df-reu 2424  df-rab 2426  df-v 2691  df-sbc 2913  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-uni 3744  df-int 3779  df-br 3937  df-opab 3997  df-tr 4034  df-exmid 4126  df-id 4222  df-iord 4295  df-on 4297  df-suc 4300  df-iom 4512  df-xp 4552  df-rel 4553  df-cnv 4554  df-co 4555  df-dm 4556  df-rn 4557  df-res 4558  df-ima 4559  df-iota 5095  df-fun 5132  df-fn 5133  df-f 5134  df-f1 5135  df-fo 5136  df-f1o 5137  df-fv 5138  df-1o 6320  df-2o 6321  df-er 6436  df-en 6642  df-fin 6644
This theorem is referenced by: (None)
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