Theorem exmidonfin 7317

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 6983 and nnon 4665. (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 2206	. . . 4 ⊢ {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}} = {{𝑥 ∈ {∅} ∣ 𝑧 = {∅}}, {𝑥 ∈ {∅} ∣ ¬ 𝑧 = {∅}}}
2	1	exmidonfinlem 7316	. . 3 ⊢ (ω = (On ∩ Fin) → DECID 𝑧 = {∅})
3	2	adantr 276	. 2 ⊢ ((ω = (On ∩ Fin) ∧ 𝑧 ⊆ {∅}) → DECID 𝑧 = {∅})
4	3	exmid1dc 4251	1 ⊢ (ω = (On ∩ Fin) → EXMID)

Syntax hints:  ¬ wn 3   → wi 4  DECID wdc 836   = wceq 1373  {crab 2489   ∩ cin 3169   ⊆ wss 3170  ∅c0 3464  {csn 3637  {cpr 3638  EXMIDwem 4245  Oncon0 4417  ωcom 4645  Fincfn 6839

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2179  ax-14 2180  ax-ext 2188  ax-sep 4169  ax-nul 4177  ax-pow 4225  ax-pr 4260  ax-un 4487  ax-setind 4592  ax-iinf 4643

This theorem depends on definitions:  df-bi 117  df-dc 837  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2193  df-cleq 2199  df-clel 2202  df-nfc 2338  df-ne 2378  df-ral 2490  df-rex 2491  df-reu 2492  df-rab 2494  df-v 2775  df-sbc 3003  df-dif 3172  df-un 3174  df-in 3176  df-ss 3183  df-nul 3465  df-pw 3622  df-sn 3643  df-pr 3644  df-op 3646  df-uni 3856  df-int 3891  df-br 4051  df-opab 4113  df-tr 4150  df-exmid 4246  df-id 4347  df-iord 4420  df-on 4422  df-suc 4425  df-iom 4646  df-xp 4688  df-rel 4689  df-cnv 4690  df-co 4691  df-dm 4692  df-rn 4693  df-res 4694  df-ima 4695  df-iota 5240  df-fun 5281  df-fn 5282  df-f 5283  df-f1 5284  df-fo 5285  df-f1o 5286  df-fv 5287  df-1o 6514  df-2o 6515  df-er 6632  df-en 6840  df-fin 6842

This theorem is referenced by: (None)