Theorem ordtriexmid 4437

Description: Ordinal trichotomy implies the law of the excluded middle (that is, decidability of an arbitrary proposition).

This theorem is stated in "Constructive ordinals", [Crosilla], p. "Set-theoretic principles incompatible with intuitionistic logic".

(Contributed by Mario Carneiro and Jim Kingdon, 14-Nov-2018.)

Hypothesis

Ref	Expression
ordtriexmid.1	⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)

Assertion

Ref	Expression
ordtriexmid	⊢ (𝜑 ∨ ¬ 𝜑)

Distinct variable groups:   𝑥,𝑦   𝜑,𝑥

Allowed substitution hint:   𝜑(𝑦)

Proof of Theorem ordtriexmid

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		noel 3367	. . . 4 ⊢ ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2		ordtriexmidlem 4435	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		eleq1 2202	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
4		eqeq1 2146	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
5		eleq2 2203	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
6	3, 4, 5	3orbi123d 1289	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
7		0elon 4314	. . . . . . . 8 ⊢ ∅ ∈ On
8		0ex 4055	. . . . . . . . 9 ⊢ ∅ ∈ V
9		eleq1 2202	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ On ↔ ∅ ∈ On))
10	9	anbi2d 459	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ ∅ ∈ On)))
11		eleq2 2203	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ∅))
12		eqeq2 2149	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 = 𝑦 ↔ 𝑥 = ∅))
13		eleq1 2202	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
14	11, 12, 13	3orbi123d 1289	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥)))
15	10, 14	imbi12d 233	. . . . . . . . 9 ⊢ (𝑦 = ∅ → (((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)) ↔ ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))))
16		ordtriexmid.1	. . . . . . . . . 10 ⊢ ∀𝑥 ∈ On ∀𝑦 ∈ On (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥)
17	16	rspec2 2521	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
18	8, 15, 17	vtocl 2740	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
19	7, 18	mpan2 421	. . . . . . 7 ⊢ (𝑥 ∈ On → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
20	6, 19	vtoclga 2752	. . . . . 6 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
21	2, 20	ax-mp 5	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
22		3orass 965	. . . . 5 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
23	21, 22	mpbi 144	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
24	1, 23	mtpor 1403	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
25		ordtriexmidlem2 4436	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
26	8	snid 3556	. . . . . 6 ⊢ ∅ ∈ {∅}
27		biidd 171	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
28	27	elrab3 2841	. . . . . 6 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
29	26, 28	ax-mp 5	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
30	29	biimpi 119	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
31	25, 30	orim12i 748	. . 3 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
32	24, 31	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
33		orcom 717	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑))
34	32, 33	mpbir 145	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 697   ∨ w3o 961   = wceq 1331   ∈ wcel 1480  ∀wral 2416  {crab 2420  ∅c0 3363  {csn 3527  Oncon0 4285

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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-nul 4054  ax-pow 4098

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-uni 3737  df-tr 4027  df-iord 4288  df-on 4290  df-suc 4293

This theorem is referenced by: (None)