Theorem exmidonfin 7150

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 6838 and nnon 4587. (Contributed by Andrew W Swan and Jim Kingdon, 9-Mar-2024.)

Assertion

Ref	Expression
exmidonfin	⊢ (ω = (On ∩ 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 2165	. . . 4 ⊢ {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}} = {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}}
2	1	exmidonfinlem 7149	. . 3 ⊢ (ω = (On ∩ Fin) → DECID 𝑧 = {∅})
3	2	adantr 274	. 2 ⊢ ((ω = (On ∩ Fin) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅})
4	3	exmid1dc 4179	1 ⊢ (ω = (On ∩ Fin) → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 824   = wceq 1343  {crab 2448   ∩ cin 3115   ⊆ wss 3116  ∅c0 3409  {csn 3576  {cpr 3577  EXMIDwem 4173  Oncon0 4341  ωcom 4567  Fincfn 6706

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-nul 4108  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-iinf 4565

This theorem depends on definitions:  df-bi 116  df-dc 825  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-ral 2449  df-rex 2450  df-reu 2451  df-rab 2453  df-v 2728  df-sbc 2952  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-nul 3410  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-int 3825  df-br 3983  df-opab 4044  df-tr 4081  df-exmid 4174  df-id 4271  df-iord 4344  df-on 4346  df-suc 4349  df-iom 4568  df-xp 4610  df-rel 4611  df-cnv 4612  df-co 4613  df-dm 4614  df-rn 4615  df-res 4616  df-ima 4617  df-iota 5153  df-fun 5190  df-fn 5191  df-f 5192  df-f1 5193  df-fo 5194  df-f1o 5195  df-fv 5196  df-1o 6384  df-2o 6385  df-er 6501  df-en 6707  df-fin 6709

This theorem is referenced by: (None)