Theorem ordtri3or 6395

Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. Theorem 1.9(iii) of [Schloeder] p. 1. (Contributed by NM, 10-May-1994.) (Proof shortened by Andrew Salmon, 25-Jul-2011.)

Assertion

Ref	Expression
ordtri3or	⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Proof of Theorem ordtri3or

Step	Hyp	Ref	Expression
1		ordin 6393	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
2		ordirr 6381	. . . . . 6 ⊢ (Ord (𝐴 ∩ 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
3	1, 2	syl 17	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
4		ianor 978	. . . . . 6 ⊢ (¬ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
5		elin 3963	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
6		incom 4200	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7	6	eleq1i 2822	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐵 ↔ (𝐵 ∩ 𝐴) ∈ 𝐵)
8	7	anbi2i 621	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
9	5, 8	bitri 274	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
10	4, 9	xchnxbir 332	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
11	3, 10	sylib 217	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
12		inss1 4227	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13		ordsseleq 6392	. . . . . . . . . 10 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴)))
14	12, 13	mpbii 232	. . . . . . . . 9 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
15	1, 14	sylan 578	. . . . . . . 8 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
16	15	anabss1 662	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
17	16	ord 860	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → (𝐴 ∩ 𝐵) = 𝐴))
18		df-ss 3964	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
19	17, 18	imbitrrdi 251	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → 𝐴 ⊆ 𝐵))
20		ordin 6393	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵 ∩ 𝐴))
21		inss1 4227	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐵
22		ordsseleq 6392	. . . . . . . . . 10 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ⊆ 𝐵 ↔ ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵)))
23	21, 22	mpbii 232	. . . . . . . . 9 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
24	20, 23	sylan 578	. . . . . . . 8 ⊢ (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
25	24	anabss4 663	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
26	25	ord 860	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → (𝐵 ∩ 𝐴) = 𝐵))
27		df-ss 3964	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
28	26, 27	imbitrrdi 251	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → 𝐵 ⊆ 𝐴))
29	19, 28	orim12d 961	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)))
30	11, 29	mpd 15	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
31		sspsstri 4101	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
32	30, 31	sylib 217	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
33		ordelpss 6391	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵))
34		biidd 261	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ 𝐴 = 𝐵))
35		ordelpss 6391	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
36	35	ancoms 457	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
37	33, 34, 36	3orbi123d 1433	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)))
38	32, 37	mpbird 256	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 394   ∨ wo 843   ∨ w3o 1084   = wceq 1539   ∈ wcel 2104   ∩ cin 3946   ⊆ wss 3947   ⊊ wpss 3948  Ord word 6362

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2701  ax-sep 5298  ax-nul 5305  ax-pr 5426

This theorem depends on definitions:  df-bi 206  df-an 395  df-or 844  df-3or 1086  df-3an 1087  df-tru 1542  df-fal 1552  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2722  df-clel 2808  df-ne 2939  df-ral 3060  df-rex 3069  df-rab 3431  df-v 3474  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-pss 3966  df-nul 4322  df-if 4528  df-pw 4603  df-sn 4628  df-pr 4630  df-op 4634  df-uni 4908  df-br 5148  df-opab 5210  df-tr 5265  df-eprel 5579  df-po 5587  df-so 5588  df-fr 5630  df-we 5632  df-ord 6366

This theorem is referenced by:  ordtri1  6396  epweon  7764  epweonALT  7765  ordeleqon  7771  poseq  8146  soseq  8147  smo11  8366  smoord  8367  omopth2  8586  ttrcltr  9713  r111  9772  tcrank  9881  domtriomlem  10439  axdc3lem2  10448  zorn2lem6  10498  grur1  10817  nosepon  27404  addsproplem7  27697  negsproplem7  27747  mulsproplem13  27823  mulsproplem14  27824  oneltri  42309