Theorem ordtri3or 6394

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 6392	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
2		ordirr 6379	. . . . . 6 ⊢ (Ord (𝐴 ∩ 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
3	1, 2	syl 18	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
4		ianor 997	. . . . . 6 ⊢ (¬ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
5		elin 3915	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
6		incom 4155	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7	6	eleq1i 2852	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐵 ↔ (𝐵 ∩ 𝐴) ∈ 𝐵)
8	7	anbi2i 635	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
9	5, 8	bitri 278	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
10	4, 9	xchnxbir 336	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
11	3, 10	sylib 221	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
12		inss1 4182	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13		ordsseleq 6391	. . . . . . . . . 10 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴)))
14	12, 13	mpbii 236	. . . . . . . . 9 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
15	1, 14	sylan 592	. . . . . . . 8 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
16	15	anabss1 679	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
17	16	ord 878	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → (𝐴 ∩ 𝐵) = 𝐴))
18		dfss2 3917	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
19	17, 18	imbitrrdi 255	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → 𝐴 ⊆ 𝐵))
20		ordin 6392	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵 ∩ 𝐴))
21		inss1 4182	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐵
22		ordsseleq 6391	. . . . . . . . . 10 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ⊆ 𝐵 ↔ ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵)))
23	21, 22	mpbii 236	. . . . . . . . 9 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
24	20, 23	sylan 592	. . . . . . . 8 ⊢ (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
25	24	anabss4 680	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
26	25	ord 878	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → (𝐵 ∩ 𝐴) = 𝐵))
27		dfss2 3917	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
28	26, 27	imbitrrdi 255	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → 𝐵 ⊆ 𝐴))
29	19, 28	orim12d 979	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴)))
30	11, 29	mpd 16	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴))
31		sspsstri 4054	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
32	30, 31	sylib 221	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
33		ordelpss 6389	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵))
34		biidd 265	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ 𝐴 = 𝐵))
35		ordelpss 6389	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
36	35	ancoms 464	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
37	33, 34, 36	3orbi123d 1463	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)))
38	32, 37	mpbird 260	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   = wceq 1570   ∈ wcel 2145   ∩ cin 3898   ⊆ wss 3899   ⊊ wpss 3900  Ord word 6360

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-ext 2733  ax-sep 5249  ax-pr 5391

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ne 2957  df-ral 3078  df-rex 3088  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-in 3906  df-ss 3916  df-pss 3919  df-nul 4280  df-if 4483  df-pw 4559  df-sn 4585  df-pr 4587  df-op 4591  df-uni 4868  df-br 5104  df-opab 5168  df-tr 5213  df-eprel 5551  df-po 5559  df-so 5560  df-fr 5604  df-we 5606  df-ord 6364

This theorem is used by:  ordtri1  6395  oneltri  6405  epweon  7787  epweonALT  7788  ordeleqon  7794  poseq  8168  soseq  8169  smo11  8365  smoord  8366  omopth2  8585  ttrcltr  9710  r111  9775  tcrank  9894  domtriomlem  10513  axdc3lem2  10522  zorn2lem6  10572  grur1  10898  nosepon  28015  addsproplem7  28354  negsproplem7  28413  mulsproplem13  28507  mulsproplem14  28508