Theorem ordtriexmid 4590

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 7392 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 3475	. . . 4 ⊢ ¬ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅
2		ordtriexmidlem 4588	. . . . . 6 ⊢ {𝑧 ∈ {∅} ∣ 𝜑} ∈ On
3		eleq1 2272	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 ∈ ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅))
4		eqeq1 2216	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (𝑥 = ∅ ↔ {𝑧 ∈ {∅} ∣ 𝜑} = ∅))
5		eleq2 2273	. . . . . . . 8 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → (∅ ∈ 𝑥 ↔ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}))
6	3, 4, 5	3orbi123d 1326	. . . . . . 7 ⊢ (𝑥 = {𝑧 ∈ {∅} ∣ 𝜑} → ((𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥) ↔ ({𝑧 ∈ {∅} ∣ 𝜑} ∈ ∅ ∨ {𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})))
7		0elon 4460	. . . . . . . 8 ⊢ ∅ ∈ On
8		0ex 4190	. . . . . . . . 9 ⊢ ∅ ∈ V
9		eleq1 2272	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ On ↔ ∅ ∈ On))
10	9	anbi2d 464	. . . . . . . . . 10 ⊢ (𝑦 = ∅ → ((𝑥 ∈ On ∧ 𝑦 ∈ On) ↔ (𝑥 ∈ On ∧ ∅ ∈ On)))
11		eleq2 2273	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 ∈ 𝑦 ↔ 𝑥 ∈ ∅))
12		eqeq2 2219	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑥 = 𝑦 ↔ 𝑥 = ∅))
13		eleq1 2272	. . . . . . . . . . 11 ⊢ (𝑦 = ∅ → (𝑦 ∈ 𝑥 ↔ ∅ ∈ 𝑥))
14	11, 12, 13	3orbi123d 1326	. . . . . . . . . 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 2599	. . . . . . . . 9 ⊢ ((𝑥 ∈ On ∧ 𝑦 ∈ On) → (𝑥 ∈ 𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦 ∈ 𝑥))
18	8, 15, 17	vtocl 2835	. . . . . . . 8 ⊢ ((𝑥 ∈ On ∧ ∅ ∈ On) → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
19	7, 18	mpan2 425	. . . . . . 7 ⊢ (𝑥 ∈ On → (𝑥 ∈ ∅ ∨ 𝑥 = ∅ ∨ ∅ ∈ 𝑥))
20	6, 19	vtoclga 2847	. . . . . 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 986	. . . . 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 1447	. . 3 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑})
25		ordtriexmidlem2 4589	. . . 4 ⊢ ({𝑧 ∈ {∅} ∣ 𝜑} = ∅ → ¬ 𝜑)
26	8	snid 3677	. . . . . 6 ⊢ ∅ ∈ {∅}
27		biidd 172	. . . . . . 7 ⊢ (𝑧 = ∅ → (𝜑 ↔ 𝜑))
28	27	elrab3 2940	. . . . . 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 763	. . 3 ⊢ (({𝑧 ∈ {∅} ∣ 𝜑} = ∅ ∨ ∅ ∈ {𝑧 ∈ {∅} ∣ 𝜑}) → (¬ 𝜑 ∨ 𝜑))
32	24, 31	ax-mp 5	. 2 ⊢ (¬ 𝜑 ∨ 𝜑)
33		orcom 732	. 2 ⊢ ((𝜑 ∨ ¬ 𝜑) ↔ (¬ 𝜑 ∨ 𝜑))
34	32, 33	mpbir 146	1 ⊢ (𝜑 ∨ ¬ 𝜑)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   ∨ wo 712   ∨ w3o 982   = wceq 1375   ∈ wcel 2180  ∀wral 2488  {crab 2492  ∅c0 3471  {csn 3646  Oncon0 4431

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-nul 4189  ax-pow 4237

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-uni 3868  df-tr 4162  df-iord 4434  df-on 4436  df-suc 4439

This theorem is referenced by: (None)