Theorem ordtriexmid 4553

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".

Also see exmidontri 7299 which is much the same theorem but biconditionalized and using the EXMID notation. (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 3450	. . . 4 ⊢ ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2		ordtriexmidlem 4551	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		eleq1 2256	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
4		eqeq1 2200	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
5		eleq2 2257	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
6	3, 4, 5	3orbi123d 1322	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
7		0elon 4423	. . . . . . . 8 ⊢ ∅ ∈ On
8		0ex 4156	. . . . . . . . 9 ⊢ ∅ ∈ V
9		eleq1 2256	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ On ↔ ∅ ∈ On))
10	9	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ ∅ ∈ On)))
11		eleq2 2257	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ∅))
12		eqeq2 2203	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 = 𝑦 ↔ 𝑥 = ∅))
13		eleq1 2256	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
14	11, 12, 13	3orbi123d 1322	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥) ↔ (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥)))
15	10, 14	imbi12d 234	. . . . . . . . 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 2583	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
18	8, 15, 17	vtocl 2814	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
19	7, 18	mpan2 425	. . . . . . 7 ⊢ (𝑥 ∈ On → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
20	6, 19	vtoclga 2826	. . . . . 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 983	. . . . 5 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
23	21, 22	mpbi 145	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
24	1, 23	mtpor 1436	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
25		ordtriexmidlem2 4552	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
26	8	snid 3649	. . . . . 6 ⊢ ∅ ∈ {∅}
27		biidd 172	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
28	27	elrab3 2917	. . . . . 6 ⊢ (∅ ∈ {∅} → (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑))
29	26, 28	ax-mp 5	. . . . 5 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} ↔ 𝜑)
30	29	biimpi 120	. . . 4 ⊢ (∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑} → 𝜑)
31	25, 30	orim12i 760	. . 3 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
32	24, 31	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
33		orcom 729	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑))
34	32, 33	mpbir 146	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∨ w3o 979   = wceq 1364   ∈ wcel 2164  ∀wral 2472  {crab 2476  ∅c0 3446  {csn 3618  Oncon0 4394

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 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-nul 4155  ax-pow 4203

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-nul 3447  df-pw 3603  df-sn 3624  df-uni 3836  df-tr 4128  df-iord 4397  df-on 4399  df-suc 4402

This theorem is referenced by: (None)