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Theorem exmidonfin 7212
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 6890 and nnon 4624. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)
Assertion
Ref Expression
exmidonfin  |-  ( om  =  ( On  i^i  Fin )  -> EXMID )

Proof of Theorem exmidonfin
Dummy variables  x  z are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 eqid 2189 . . . 4  |-  { {
x  e.  { (/) }  |  z  =  { (/)
} } ,  {
x  e.  { (/) }  |  -.  z  =  { (/) } } }  =  { { x  e. 
{ (/) }  |  z  =  { (/) } } ,  { x  e.  { (/)
}  |  -.  z  =  { (/) } } }
21exmidonfinlem 7211 . . 3  |-  ( om  =  ( On  i^i  Fin )  -> DECID  z  =  { (/)
} )
32adantr 276 . 2  |-  ( ( om  =  ( On 
i^i  Fin )  /\  z  C_ 
{ (/) } )  -> DECID  z  =  { (/) } )
43exmid1dc 4215 1  |-  ( om  =  ( On  i^i  Fin )  -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 835    = wceq 1364   {crab 2472    i^i cin 3143    C_ wss 3144   (/)c0 3437   {csn 3607   {cpr 3608  EXMIDwem 4209   Oncon0 4378   omcom 4604   Fincfn 6758
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 4136  ax-nul 4144  ax-pow 4189  ax-pr 4224  ax-un 4448  ax-setind 4551  ax-iinf 4602
This theorem depends on definitions:  df-bi 117  df-dc 836  df-3or 981  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-ne 2361  df-ral 2473  df-rex 2474  df-reu 2475  df-rab 2477  df-v 2754  df-sbc 2978  df-dif 3146  df-un 3148  df-in 3150  df-ss 3157  df-nul 3438  df-pw 3592  df-sn 3613  df-pr 3614  df-op 3616  df-uni 3825  df-int 3860  df-br 4019  df-opab 4080  df-tr 4117  df-exmid 4210  df-id 4308  df-iord 4381  df-on 4383  df-suc 4386  df-iom 4605  df-xp 4647  df-rel 4648  df-cnv 4649  df-co 4650  df-dm 4651  df-rn 4652  df-res 4653  df-ima 4654  df-iota 5193  df-fun 5233  df-fn 5234  df-f 5235  df-f1 5236  df-fo 5237  df-f1o 5238  df-fv 5239  df-1o 6435  df-2o 6436  df-er 6553  df-en 6759  df-fin 6761
This theorem is referenced by: (None)
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