Theorem exmidonfin 7224

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 6901 and nnon 4627. (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.

Step	Hyp	Ref	Expression
1		eqid 2189	. . . 4 |- { { x e. { (/) } | z = { (/) } } , { x e. { (/) } | -. z = { (/) } } } = { { x e. { (/) } | z = { (/) } } , { x e. { (/) } | -. z = { (/) } } }
2	1	exmidonfinlem 7223	. . 3 |- ( om = ( On i^i Fin ) -> DECID z = { (/) } )
3	2	adantr 276	. 2 |- ( ( om = ( On i^i Fin ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
4	3	exmid1dc 4218	1 |- ( om = ( On i^i Fin ) -> EXMID )

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 4212   Oncon0 4381   omcom 4607   Fincfn 6767

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 4192  ax-pr 4227  ax-un 4451  ax-setind 4554  ax-iinf 4605

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 4213  df-id 4311  df-iord 4384  df-on 4386  df-suc 4389  df-iom 4608  df-xp 4650  df-rel 4651  df-cnv 4652  df-co 4653  df-dm 4654  df-rn 4655  df-res 4656  df-ima 4657  df-iota 5196  df-fun 5237  df-fn 5238  df-f 5239  df-f1 5240  df-fo 5241  df-f1o 5242  df-fv 5243  df-1o 6442  df-2o 6443  df-er 6560  df-en 6768  df-fin 6770

This theorem is referenced by: (None)