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Theorem exmidonfin 7055
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 6766 and nnon 4523. (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 2139 . . . 4  |-  { {
x  e.  { (/) }  |  z  =  { (/)
} } ,  {
x  e.  { (/) }  |  -.  z  =  { (/) } } }  =  { { x  e. 
{ (/) }  |  z  =  { (/) } } ,  { x  e.  { (/)
}  |  -.  z  =  { (/) } } }
21exmidonfinlem 7054 . . 3  |-  ( om  =  ( On  i^i  Fin )  -> DECID  z  =  { (/)
} )
32adantr 274 . 2  |-  ( ( om  =  ( On 
i^i  Fin )  /\  z  C_ 
{ (/) } )  -> DECID  z  =  { (/) } )
43exmid1dc 4123 1  |-  ( om  =  ( On  i^i  Fin )  -> EXMID )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4  DECID wdc 819    = wceq 1331   {crab 2420    i^i cin 3070    C_ wss 3071   (/)c0 3363   {csn 3527   {cpr 3528  EXMIDwem 4118   Oncon0 4285   omcom 4504   Fincfn 6634
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-dc 820  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-br 3930  df-opab 3990  df-tr 4027  df-exmid 4119  df-id 4215  df-iord 4288  df-on 4290  df-suc 4293  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1o 6313  df-2o 6314  df-er 6429  df-en 6635  df-fin 6637
This theorem is referenced by: (None)
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