Theorem ordtri3or 5755

Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38. (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 5753	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → Ord (𝐴 ∩ 𝐵))
2		ordirr 5741	. . . . . 6 ⊢ (Ord (𝐴 ∩ 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
3	1, 2	syl 17	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵))
4		ianor 509	. . . . . 6 ⊢ (¬ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
5		elin 3796	. . . . . . 7 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵))
6		incom 3805	. . . . . . . . 9 ⊢ (𝐴 ∩ 𝐵) = (𝐵 ∩ 𝐴)
7	6	eleq1i 2692	. . . . . . . 8 ⊢ ((𝐴 ∩ 𝐵) ∈ 𝐵 ↔ (𝐵 ∩ 𝐴) ∈ 𝐵)
8	7	anbi2i 730	. . . . . . 7 ⊢ (((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐴 ∩ 𝐵) ∈ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
9	5, 8	bitri 264	. . . . . 6 ⊢ ((𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∧ (𝐵 ∩ 𝐴) ∈ 𝐵))
10	4, 9	xchnxbir 323	. . . . 5 ⊢ (¬ (𝐴 ∩ 𝐵) ∈ (𝐴 ∩ 𝐵) ↔ (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
11	3, 10	sylib 208	. . . 4 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 ∨ ¬ (𝐵 ∩ 𝐴) ∈ 𝐵))
12		inss1 3833	. . . . . . . . . 10 ⊢ (𝐴 ∩ 𝐵) ⊆ 𝐴
13		ordsseleq 5752	. . . . . . . . . 10 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ⊆ 𝐴 ↔ ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴)))
14	12, 13	mpbii 223	. . . . . . . . 9 ⊢ ((Ord (𝐴 ∩ 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
15	1, 14	sylan 488	. . . . . . . 8 ⊢ (((Ord 𝐴 ∧ Ord 𝐵) ∧ Ord 𝐴) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
16	15	anabss1 855	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∩ 𝐵) ∈ 𝐴 ∨ (𝐴 ∩ 𝐵) = 𝐴))
17	16	ord 392	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → (𝐴 ∩ 𝐵) = 𝐴))
18		df-ss 3588	. . . . . 6 ⊢ (𝐴 ⊆ 𝐵 ↔ (𝐴 ∩ 𝐵) = 𝐴)
19	17, 18	syl6ibr 242	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐴 ∩ 𝐵) ∈ 𝐴 → 𝐴 ⊆ 𝐵))
20		ordin 5753	. . . . . . . . 9 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → Ord (𝐵 ∩ 𝐴))
21		inss1 3833	. . . . . . . . . 10 ⊢ (𝐵 ∩ 𝐴) ⊆ 𝐵
22		ordsseleq 5752	. . . . . . . . . 10 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ⊆ 𝐵 ↔ ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵)))
23	21, 22	mpbii 223	. . . . . . . . 9 ⊢ ((Ord (𝐵 ∩ 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
24	20, 23	sylan 488	. . . . . . . 8 ⊢ (((Ord 𝐵 ∧ Ord 𝐴) ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
25	24	anabss4 856	. . . . . . 7 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐵 ∩ 𝐴) ∈ 𝐵 ∨ (𝐵 ∩ 𝐴) = 𝐵))
26	25	ord 392	. . . . . 6 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → (𝐵 ∩ 𝐴) = 𝐵))
27		df-ss 3588	. . . . . 6 ⊢ (𝐵 ⊆ 𝐴 ↔ (𝐵 ∩ 𝐴) = 𝐵)
28	26, 27	syl6ibr 242	. . . . 5 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (¬ (𝐵 ∩ 𝐴) ∈ 𝐵 → 𝐵 ⊆ 𝐴))
29	19, 28	orim12d 883	. . . 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 3709	. . 3 ⊢ ((𝐴 ⊆ 𝐵 ∨ 𝐵 ⊆ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
32	30, 31	sylib 208	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴))
33		ordelpss 5751	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ↔ 𝐴 ⊊ 𝐵))
34		biidd 252	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 = 𝐵 ↔ 𝐴 = 𝐵))
35		ordelpss 5751	. . . 4 ⊢ ((Ord 𝐵 ∧ Ord 𝐴) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
36	35	ancoms 469	. . 3 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐵 ∈ 𝐴 ↔ 𝐵 ⊊ 𝐴))
37	33, 34, 36	3orbi123d 1398	. 2 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → ((𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴) ↔ (𝐴 ⊊ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ⊊ 𝐴)))
38	32, 37	mpbird 247	1 ⊢ ((Ord 𝐴 ∧ Ord 𝐵) → (𝐴 ∈ 𝐵 ∨ 𝐴 = 𝐵 ∨ 𝐵 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 196   ∨ wo 383   ∧ wa 384   ∨ w3o 1036   = wceq 1483   ∈ wcel 1990   ∩ cin 3573   ⊆ wss 3574   ⊊ wpss 3575  Ord word 5722

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602  ax-sep 4781  ax-nul 4789  ax-pr 4906

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3or 1038  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-eu 2474  df-mo 2475  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-ne 2795  df-ral 2917  df-rex 2918  df-rab 2921  df-v 3202  df-sbc 3436  df-dif 3577  df-un 3579  df-in 3581  df-ss 3588  df-pss 3590  df-nul 3916  df-if 4087  df-sn 4178  df-pr 4180  df-op 4184  df-uni 4437  df-br 4654  df-opab 4713  df-tr 4753  df-eprel 5029  df-po 5035  df-so 5036  df-fr 5073  df-we 5075  df-ord 5726

This theorem is referenced by:  ordtri1  5756  ordtri3OLD  5760  ordon  6982  ordeleqon  6988  smo11  7461  smoord  7462  omopth2  7664  r111  8638  tcrank  8747  domtriomlem  9264  axdc3lem2  9273  zorn2lem6  9323  grur1  9642  poseq  31750  soseq  31751  nosepon  31818