Theorem exmidonfin 7404

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 7058 and nnon 4708. (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 2231	. . . 4 ⊢ {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}} = {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}}
2	1	exmidonfinlem 7403	. . 3 ⊢ (ω = (On ∩ Fin) → DECID 𝑧 = {∅})
3	2	adantr 276	. 2 ⊢ ((ω = (On ∩ Fin) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅})
4	3	exmid1dc 4290	1 ⊢ (ω = (On ∩ Fin) → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 841   = wceq 1397  {crab 2514   ∩ cin 3199   ⊆ wss 3200  ∅c0 3494  {csn 3669  {cpr 3670  EXMIDwem 4284  Oncon0 4460  ωcom 4688  Fincfn 6908

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 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-13 2204  ax-14 2205  ax-ext 2213  ax-sep 4207  ax-nul 4215  ax-pow 4264  ax-pr 4299  ax-un 4530  ax-setind 4635  ax-iinf 4686

This theorem depends on definitions:  df-bi 117  df-dc 842  df-3or 1005  df-3an 1006  df-tru 1400  df-fal 1403  df-nf 1509  df-sb 1811  df-eu 2082  df-mo 2083  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ne 2403  df-ral 2515  df-rex 2516  df-reu 2517  df-rab 2519  df-v 2804  df-sbc 3032  df-dif 3202  df-un 3204  df-in 3206  df-ss 3213  df-nul 3495  df-pw 3654  df-sn 3675  df-pr 3676  df-op 3678  df-uni 3894  df-int 3929  df-br 4089  df-opab 4151  df-tr 4188  df-exmid 4285  df-id 4390  df-iord 4463  df-on 4465  df-suc 4468  df-iom 4689  df-xp 4731  df-rel 4732  df-cnv 4733  df-co 4734  df-dm 4735  df-rn 4736  df-res 4737  df-ima 4738  df-iota 5286  df-fun 5328  df-fn 5329  df-f 5330  df-f1 5331  df-fo 5332  df-f1o 5333  df-fv 5334  df-1o 6581  df-2o 6582  df-er 6701  df-en 6909  df-fin 6911

This theorem is referenced by: (None)