Theorem ordtri3or 6366

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 6364	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
2		ordirr 6352	. . . . . 6 ⊢ (Ord (𝐴 ∩ 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
3	1, 2	syl 17	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
4		ianor 983	. . . . . 6 ⊢ (¬ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
5		elin 3932	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
6		incom 4174	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7	6	eleq1i 2820	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐵 ↔ (𝐵 ∩ 𝐴) ∈ 𝐵)
8	7	anbi2i 623	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
9	5, 8	bitri 275	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
10	4, 9	xchnxbir 333	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
11	3, 10	sylib 218	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
12		inss1 4202	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13		ordsseleq 6363	. . . . . . . . . 10 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴)))
14	12, 13	mpbii 233	. . . . . . . . 9 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
15	1, 14	sylan 580	. . . . . . . 8 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
16	15	anabss1 666	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
17	16	ord 864	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → (𝐴 ∩ 𝐵) = 𝐴))
18		dfss2 3934	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
19	17, 18	imbitrrdi 252	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → 𝐴 ⊆ 𝐵))
20		ordin 6364	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵 ∩ 𝐴))
21		inss1 4202	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐵
22		ordsseleq 6363	. . . . . . . . . 10 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ⊆ 𝐵 ↔ ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵)))
23	21, 22	mpbii 233	. . . . . . . . 9 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
24	20, 23	sylan 580	. . . . . . . 8 ⊢ (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
25	24	anabss4 667	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
26	25	ord 864	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → (𝐵 ∩ 𝐴) = 𝐵))
27		dfss2 3934	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
28	26, 27	imbitrrdi 252	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → 𝐵 ⊆ 𝐴))
29	19, 28	orim12d 966	. . . 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 4070	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
32	30, 31	sylib 218	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
33		ordelpss 6362	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵))
34		biidd 262	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ 𝐴 = 𝐵))
35		ordelpss 6362	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
36	35	ancoms 458	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
37	33, 34, 36	3orbi123d 1437	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)))
38	32, 37	mpbird 257	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1540   ∈ wcel 2109   ∩ cin 3915   ⊆ wss 3916   ⊊ wpss 3917  Ord word 6333

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 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2702  ax-sep 5253  ax-nul 5263  ax-pr 5389

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-ne 2927  df-ral 3046  df-rex 3055  df-rab 3409  df-v 3452  df-dif 3919  df-un 3921  df-in 3923  df-ss 3933  df-pss 3936  df-nul 4299  df-if 4491  df-pw 4567  df-sn 4592  df-pr 4594  df-op 4598  df-uni 4874  df-br 5110  df-opab 5172  df-tr 5217  df-eprel 5540  df-po 5548  df-so 5549  df-fr 5593  df-we 5595  df-ord 6337

This theorem is referenced by:  ordtri1  6367  oneltri  6377  epweon  7753  epweonALT  7754  ordeleqon  7760  poseq  8139  soseq  8140  smo11  8335  smoord  8336  omopth2  8550  ttrcltr  9675  r111  9734  tcrank  9843  domtriomlem  10401  axdc3lem2  10410  zorn2lem6  10460  grur1  10779  nosepon  27583  addsproplem7  27888  negsproplem7  27946  mulsproplem13  28037  mulsproplem14  28038