Theorem ordtri3or 2230

Description: A trichotomy law for ordinals. Proposition 7.10 of [TakeutiZaring] p. 38.

Assertion

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

Proof of Theorem ordtri3or

Step	Hyp	Ref	Expression
1		ordin 2228	. . . . . 6 ⊢ ((Ord A ∧ Ord B) → Ord (A ∩ B))
2		ordeirr 2217	. . . . . 6 ⊢ (Ord (A ∩ B) → ¬ (A ∩ B) ∈ (A ∩ B))
3	1, 2	syl 12	. . . . 5 ⊢ ((Ord A ∧ Ord B) → ¬ (A ∩ B) ∈ (A ∩ B))
4		elin 1635	. . . . . . . 8 ⊢ ((A ∩ B) ∈ (A ∩ B) ↔ ((A ∩ B) ∈ A ∧ (A ∩ B) ∈ B))
5		incom 1636	. . . . . . . . . 10 ⊢ (A ∩ B) = (B ∩ A)
6	5	eleq1i 1152	. . . . . . . . 9 ⊢ ((A ∩ B) ∈ B ↔ (B ∩ A) ∈ B)
7	6	anbi2i 367	. . . . . . . 8 ⊢ (((A ∩ B) ∈ A ∧ (A ∩ B) ∈ B) ↔ ((A ∩ B) ∈ A ∧ (B ∩ A) ∈ B))
8	4, 7	bitr 151	. . . . . . 7 ⊢ ((A ∩ B) ∈ (A ∩ B) ↔ ((A ∩ B) ∈ A ∧ (B ∩ A) ∈ B))
9	8	negbii 162	. . . . . 6 ⊢ (¬ (A ∩ B) ∈ (A ∩ B) ↔ ¬ ((A ∩ B) ∈ A ∧ (B ∩ A) ∈ B))
10		ianor 253	. . . . . 6 ⊢ (¬ ((A ∩ B) ∈ A ∧ (B ∩ A) ∈ B) ↔ (¬ (A ∩ B) ∈ A ∨ ¬ (B ∩ A) ∈ B))
11	9, 10	bitr 151	. . . . 5 ⊢ (¬ (A ∩ B) ∈ (A ∩ B) ↔ (¬ (A ∩ B) ∈ A ∨ ¬ (B ∩ A) ∈ B))
12	3, 11	sylib 173	. . . 4 ⊢ ((Ord A ∧ Ord B) → (¬ (A ∩ B) ∈ A ∨ ¬ (B ∩ A) ∈ B))
13		inss1 1657	. . . . . . . . . 10 ⊢ (A ∩ B) ⊆ A
14		ordsseleq 2227	. . . . . . . . . 10 ⊢ ((Ord (A ∩ B) ∧ Ord A) → ((A ∩ B) ⊆ A ↔ ((A ∩ B) ∈ A ∨ (A ∩ B) = A)))
15	13, 14	mpbii 168	. . . . . . . . 9 ⊢ ((Ord (A ∩ B) ∧ Ord A) → ((A ∩ B) ∈ A ∨ (A ∩ B) = A))
16	15, 1	sylan 343	. . . . . . . 8 ⊢ (((Ord A ∧ Ord B) ∧ Ord A) → ((A ∩ B) ∈ A ∨ (A ∩ B) = A))
17	16	anabss1 381	. . . . . . 7 ⊢ ((Ord A ∧ Ord B) → ((A ∩ B) ∈ A ∨ (A ∩ B) = A))
18	17	ord 202	. . . . . 6 ⊢ ((Ord A ∧ Ord B) → (¬ (A ∩ B) ∈ A → (A ∩ B) = A))
19		df-ss 1492	. . . . . 6 ⊢ (A ⊆ B ↔ (A ∩ B) = A)
20	18, 19	syl6ibr 186	. . . . 5 ⊢ ((Ord A ∧ Ord B) → (¬ (A ∩ B) ∈ A → A ⊆ B))
21		inss1 1657	. . . . . . . . . 10 ⊢ (B ∩ A) ⊆ B
22		ordsseleq 2227	. . . . . . . . . 10 ⊢ ((Ord (B ∩ A) ∧ Ord B) → ((B ∩ A) ⊆ B ↔ ((B ∩ A) ∈ B ∨ (B ∩ A) = B)))
23	21, 22	mpbii 168	. . . . . . . . 9 ⊢ ((Ord (B ∩ A) ∧ Ord B) → ((B ∩ A) ∈ B ∨ (B ∩ A) = B))
24		ordin 2228	. . . . . . . . 9 ⊢ ((Ord B ∧ Ord A) → Ord (B ∩ A))
25	23, 24	sylan 343	. . . . . . . 8 ⊢ (((Ord B ∧ Ord A) ∧ Ord B) → ((B ∩ A) ∈ B ∨ (B ∩ A) = B))
26	25	anabss4 383	. . . . . . 7 ⊢ ((Ord A ∧ Ord B) → ((B ∩ A) ∈ B ∨ (B ∩ A) = B))
27	26	ord 202	. . . . . 6 ⊢ ((Ord A ∧ Ord B) → (¬ (B ∩ A) ∈ B → (B ∩ A) = B))
28		df-ss 1492	. . . . . 6 ⊢ (B ⊆ A ↔ (B ∩ A) = B)
29	27, 28	syl6ibr 186	. . . . 5 ⊢ ((Ord A ∧ Ord B) → (¬ (B ∩ A) ∈ B → B ⊆ A))
30	20, 29	orim12d 436	. . . 4 ⊢ ((Ord A ∧ Ord B) → ((¬ (A ∩ B) ∈ A ∨ ¬ (B ∩ A) ∈ B) → (A ⊆ B ∨ B ⊆ A)))
31	12, 30	mpd 46	. . 3 ⊢ ((Ord A ∧ Ord B) → (A ⊆ B ∨ B ⊆ A))
32		ordsseleq 2227	. . . 4 ⊢ ((Ord A ∧ Ord B) → (A ⊆ B ↔ (A ∈ B ∨ A = B)))
33		ordsseleq 2227	. . . . 5 ⊢ ((Ord B ∧ Ord A) → (B ⊆ A ↔ (B ∈ A ∨ B = A)))
34	33	ancoms 334	. . . 4 ⊢ ((Ord A ∧ Ord B) → (B ⊆ A ↔ (B ∈ A ∨ B = A)))
35	32, 34	orbi12d 475	. . 3 ⊢ ((Ord A ∧ Ord B) → ((A ⊆ B ∨ B ⊆ A) ↔ ((A ∈ B ∨ A = B) ∨ (B ∈ A ∨ B = A))))
36	31, 35	mpbid 170	. 2 ⊢ ((Ord A ∧ Ord B) → ((A ∈ B ∨ A = B) ∨ (B ∈ A ∨ B = A)))
37		df-3or 582	. . 3 ⊢ ((A ∈ B ∨ A = B ∨ B ∈ A) ↔ ((A ∈ B ∨ A = B) ∨ B ∈ A))
38		oridm 208	. . . . . 6 ⊢ ((A = B ∨ A = B) ↔ A = B)
39		cleqcom 1103	. . . . . . 7 ⊢ (A = B ↔ B = A)
40	39	orbi2i 214	. . . . . 6 ⊢ ((A = B ∨ A = B) ↔ (A = B ∨ B = A))
41	38, 40	bitr3 153	. . . . 5 ⊢ (A = B ↔ (A = B ∨ B = A))
42	41	orbi2i 214	. . . 4 ⊢ (((A ∈ B ∨ B ∈ A) ∨ A = B) ↔ ((A ∈ B ∨ B ∈ A) ∨ (A = B ∨ B = A)))
43		or23 219	. . . 4 ⊢ (((A ∈ B ∨ A = B) ∨ B ∈ A) ↔ ((A ∈ B ∨ B ∈ A) ∨ A = B))
44		or4 220	. . . 4 ⊢ (((A ∈ B ∨ A = B) ∨ (B ∈ A ∨ B = A)) ↔ ((A ∈ B ∨ B ∈ A) ∨ (A = B ∨ B = A)))
45	42, 43, 44	3bitr4 158	. . 3 ⊢ (((A ∈ B ∨ A = B) ∨ B ∈ A) ↔ ((A ∈ B ∨ A = B) ∨ (B ∈ A ∨ B = A)))
46	37, 45	bitr 151	. 2 ⊢ ((A ∈ B ∨ A = B ∨ B ∈ A) ↔ ((A ∈ B ∨ A = B) ∨ (B ∈ A ∨ B = A)))
47	36, 46	sylibr 175	1 ⊢ ((Ord A ∧ Ord B) → (A ∈ B ∨ A = B ∨ B ∈ A))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   ∨ w3o 580   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487  Ord word 2198

This theorem is referenced by:  ordtri1 2231  ordtri3 2234  ordon 2238  ordeleqon 2241  ordtri2or 2328  zornlem6 3608

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-rex 1206  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-uni 1920  df-tr 2042  df-br 2063  df-opab 2098  df-eprel 2122  df-po 2128  df-so 2138  df-fr 2169  df-we 2186  df-ord 2202

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