Theorem exmidonfin 7372

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 7034 and nnon 4702. (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 2229	. . . 4 |- { { x e. { (/) } | z = { (/) } } , { x e. { (/) } | -. z = { (/) } } } = { { x e. { (/) } | z = { (/) } } , { x e. { (/) } | -. z = { (/) } } }
2	1	exmidonfinlem 7371	. . 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 4284	1 |- ( om = ( On i^i Fin ) -> EXMID )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 839   = wceq 1395   {crab 2512   i^i cin 3196   C_ wss 3197   (/)c0 3491   {csn 3666   {cpr 3667  EXMIDwem 4278   Oncon0 4454   omcom 4682   Fincfn 6887

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-nul 4210  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-iinf 4680

This theorem depends on definitions:  df-bi 117  df-dc 840  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-ral 2513  df-rex 2514  df-reu 2515  df-rab 2517  df-v 2801  df-sbc 3029  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-nul 3492  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-int 3924  df-br 4084  df-opab 4146  df-tr 4183  df-exmid 4279  df-id 4384  df-iord 4457  df-on 4459  df-suc 4462  df-iom 4683  df-xp 4725  df-rel 4726  df-cnv 4727  df-co 4728  df-dm 4729  df-rn 4730  df-res 4731  df-ima 4732  df-iota 5278  df-fun 5320  df-fn 5321  df-f 5322  df-f1 5323  df-fo 5324  df-f1o 5325  df-fv 5326  df-1o 6562  df-2o 6563  df-er 6680  df-en 6888  df-fin 6890

This theorem is referenced by: (None)