Theorem ordtriexmid 4247

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 3228	. . . 4 ⊢ ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2		ordtriexmidlem 4245	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		eleq1 2100	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
4		eqeq1 2046	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
5		eleq2 2101	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
6	3, 4, 5	3orbi123d 1206	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
7		0elon 4129	. . . . . . . 8 ⊢ ∅ ∈ On
8		0ex 3884	. . . . . . . . 9 ⊢ ∅ ∈ V
9		eleq1 2100	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ On ↔ ∅ ∈ On))
10	9	anbi2d 437	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ ∅ ∈ On)))
11		eleq2 2101	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ∅))
12		eqeq2 2049	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 = 𝑦 ↔ 𝑥 = ∅))
13		eleq1 2100	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
14	11, 12, 13	3orbi123d 1206	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥)))
15	10, 14	imbi12d 223	. . . . . . . . 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 2408	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
18	8, 15, 17	vtocl 2608	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
19	7, 18	mpan2 401	. . . . . . 7 ⊢ (𝑥 ∈ On → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
20	6, 19	vtoclga 2619	. . . . . 6 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ On → ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
21	2, 20	ax-mp 7	. . . . 5 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
22		3orass 888	. . . . 5 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
23	21, 22	mpbi 133	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
24	1, 23	mtpor 1316	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
25		ordtriexmidlem2 4246	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
26	8	snid 3402	. . . . . 6 ⊢ ∅ ∈ {∅}
27		biidd 161	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
28	27	elrab3 2699	. . . . . 6 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
29	26, 28	ax-mp 7	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
30	29	biimpi 113	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
31	25, 30	orim12i 676	. . 3 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
32	24, 31	ax-mp 7	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
33		orcom 647	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑))
34	32, 33	mpbir 134	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 629   ∨ w3o 884   = wceq 1243   ∈ wcel 1393  ∀wral 2306  {crab 2310  ∅c0 3224  {csn 3375  Oncon0 4100

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-nul 3883  ax-pow 3927

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-uni 3581  df-tr 3855  df-iord 4103  df-on 4105  df-suc 4108

This theorem is referenced by: (None)