Theorem exmidonfin 7171

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 6850 and nnon 4594. (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 2170	. . . 4 |- { { x e. { (/) } | z = { (/) } } , { x e. { (/) } | -. z = { (/) } } } = { { x e. { (/) } | z = { (/) } } , { x e. { (/) } | -. z = { (/) } } }
2	1	exmidonfinlem 7170	. . 3 |- ( om = ( On i^i Fin ) -> DECID z = { (/) } )
3	2	adantr 274	. 2 |- ( ( om = ( On i^i Fin ) /\ z C_ { (/) } ) -> DECID z = { (/) } )
4	3	exmid1dc 4186	1 |- ( om = ( On i^i Fin ) -> EXMID )

Syntax hints:   -. wn 3   -> wi 4  DECID wdc 829   = wceq 1348   {crab 2452   i^i cin 3120   C_ wss 3121   (/)c0 3414   {csn 3583   {cpr 3584  EXMIDwem 4180   Oncon0 4348   omcom 4574   Fincfn 6718

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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572

This theorem depends on definitions:  df-bi 116  df-dc 830  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-int 3832  df-br 3990  df-opab 4051  df-tr 4088  df-exmid 4181  df-id 4278  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-1o 6395  df-2o 6396  df-er 6513  df-en 6719  df-fin 6721

This theorem is referenced by: (None)