Theorem ordtri3or 2979

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 2977	. . . . . 6 ⊢ ((Ord A ⋀ Ord B) → Ord (A ∩ B))
2		ordirr 2966	. . . . . 6 ⊢ (Ord (A ∩ B) → ¬ (A ∩ B) ∈ (A ∩ B))
3	1, 2	syl 10	. . . . 5 ⊢ ((Ord A ⋀ Ord B) → ¬ (A ∩ B) ∈ (A ∩ B))
4		elin 2207	. . . . . . . 8 ⊢ ((A ∩ B) ∈ (A ∩ B) ↔ ((A ∩ B) ∈ A ⋀ (A ∩ B) ∈ B))
5		incom 2208	. . . . . . . . . 10 ⊢ (A ∩ B) = (B ∩ A)
6	5	eleq1i 1537	. . . . . . . . 9 ⊢ ((A ∩ B) ∈ B ↔ (B ∩ A) ∈ B)
7	6	anbi2i 480	. . . . . . . 8 ⊢ (((A ∩ B) ∈ A ⋀ (A ∩ B) ∈ B) ↔ ((A ∩ B) ∈ A ⋀ (B ∩ A) ∈ B))
8	4, 7	bitr 173	. . . . . . 7 ⊢ ((A ∩ B) ∈ (A ∩ B) ↔ ((A ∩ B) ∈ A ⋀ (B ∩ A) ∈ B))
9	8	negbii 187	. . . . . 6 ⊢ (¬ (A ∩ B) ∈ (A ∩ B) ↔ ¬ ((A ∩ B) ∈ A ⋀ (B ∩ A) ∈ B))
10		ianor 305	. . . . . 6 ⊢ (¬ ((A ∩ B) ∈ A ⋀ (B ∩ A) ∈ B) ↔ (¬ (A ∩ B) ∈ A ⋁ ¬ (B ∩ A) ∈ B))
11	9, 10	bitr 173	. . . . 5 ⊢ (¬ (A ∩ B) ∈ (A ∩ B) ↔ (¬ (A ∩ B) ∈ A ⋁ ¬ (B ∩ A) ∈ B))
12	3, 11	sylib 198	. . . 4 ⊢ ((Ord A ⋀ Ord B) → (¬ (A ∩ B) ∈ A ⋁ ¬ (B ∩ A) ∈ B))
13		inss1 2230	. . . . . . . . . 10 ⊢ (A ∩ B) ⊆ A
14		ordsseleq 2976	. . . . . . . . . 10 ⊢ ((Ord (A ∩ B) ⋀ Ord A) → ((A ∩ B) ⊆ A ↔ ((A ∩ B) ∈ A ⋁ (A ∩ B) = A)))
15	13, 14	mpbii 193	. . . . . . . . 9 ⊢ ((Ord (A ∩ B) ⋀ Ord A) → ((A ∩ B) ∈ A ⋁ (A ∩ B) = A))
16	15, 1	sylan 448	. . . . . . . 8 ⊢ (((Ord A ⋀ Ord B) ⋀ Ord A) → ((A ∩ B) ∈ A ⋁ (A ∩ B) = A))
17	16	anabss1 499	. . . . . . 7 ⊢ ((Ord A ⋀ Ord B) → ((A ∩ B) ∈ A ⋁ (A ∩ B) = A))
18	17	ord 232	. . . . . 6 ⊢ ((Ord A ⋀ Ord B) → (¬ (A ∩ B) ∈ A → (A ∩ B) = A))
19		df-ss 2053	. . . . . 6 ⊢ (A ⊆ B ↔ (A ∩ B) = A)
20	18, 19	syl6ibr 213	. . . . 5 ⊢ ((Ord A ⋀ Ord B) → (¬ (A ∩ B) ∈ A → A ⊆ B))
21		inss1 2230	. . . . . . . . . 10 ⊢ (B ∩ A) ⊆ B
22		ordsseleq 2976	. . . . . . . . . 10 ⊢ ((Ord (B ∩ A) ⋀ Ord B) → ((B ∩ A) ⊆ B ↔ ((B ∩ A) ∈ B ⋁ (B ∩ A) = B)))
23	21, 22	mpbii 193	. . . . . . . . 9 ⊢ ((Ord (B ∩ A) ⋀ Ord B) → ((B ∩ A) ∈ B ⋁ (B ∩ A) = B))
24		ordin 2977	. . . . . . . . 9 ⊢ ((Ord B ⋀ Ord A) → Ord (B ∩ A))
25	23, 24	sylan 448	. . . . . . . 8 ⊢ (((Ord B ⋀ Ord A) ⋀ Ord B) → ((B ∩ A) ∈ B ⋁ (B ∩ A) = B))
26	25	anabss4 501	. . . . . . 7 ⊢ ((Ord A ⋀ Ord B) → ((B ∩ A) ∈ B ⋁ (B ∩ A) = B))
27	26	ord 232	. . . . . 6 ⊢ ((Ord A ⋀ Ord B) → (¬ (B ∩ A) ∈ B → (B ∩ A) = B))
28		df-ss 2053	. . . . . 6 ⊢ (B ⊆ A ↔ (B ∩ A) = B)
29	27, 28	syl6ibr 213	. . . . 5 ⊢ ((Ord A ⋀ Ord B) → (¬ (B ∩ A) ∈ B → B ⊆ A))
30	20, 29	orim12d 565	. . . 4 ⊢ ((Ord A ⋀ Ord B) → ((¬ (A ∩ B) ∈ A ⋁ ¬ (B ∩ A) ∈ B) → (A ⊆ B ⋁ B ⊆ A)))
31	12, 30	mpd 26	. . 3 ⊢ ((Ord A ⋀ Ord B) → (A ⊆ B ⋁ B ⊆ A))
32		ordsseleq 2976	. . . 4 ⊢ ((Ord A ⋀ Ord B) → (A ⊆ B ↔ (A ∈ B ⋁ A = B)))
33		ordsseleq 2976	. . . . 5 ⊢ ((Ord B ⋀ Ord A) → (B ⊆ A ↔ (B ∈ A ⋁ B = A)))
34	33	ancoms 436	. . . 4 ⊢ ((Ord A ⋀ Ord B) → (B ⊆ A ↔ (B ∈ A ⋁ B = A)))
35	32, 34	orbi12d 627	. . 3 ⊢ ((Ord A ⋀ Ord B) → ((A ⊆ B ⋁ B ⊆ A) ↔ ((A ∈ B ⋁ A = B) ⋁ (B ∈ A ⋁ B = A))))
36	31, 35	mpbid 195	. 2 ⊢ ((Ord A ⋀ Ord B) → ((A ∈ B ⋁ A = B) ⋁ (B ∈ A ⋁ B = A)))
37		df-3or 776	. . . 4 ⊢ ((A ∈ B ⋁ A = B ⋁ B ∈ A) ↔ ((A ∈ B ⋁ A = B) ⋁ B ∈ A))
38		or23 263	. . . 4 ⊢ (((A ∈ B ⋁ A = B) ⋁ B ∈ A) ↔ ((A ∈ B ⋁ B ∈ A) ⋁ A = B))
39	37, 38	bitr 173	. . 3 ⊢ ((A ∈ B ⋁ A = B ⋁ B ∈ A) ↔ ((A ∈ B ⋁ B ∈ A) ⋁ A = B))
40		orordir 267	. . 3 ⊢ (((A ∈ B ⋁ B ∈ A) ⋁ A = B) ↔ ((A ∈ B ⋁ A = B) ⋁ (B ∈ A ⋁ A = B)))
41		eqcom 1477	. . . . 5 ⊢ (A = B ↔ B = A)
42	41	orbi2i 255	. . . 4 ⊢ ((B ∈ A ⋁ A = B) ↔ (B ∈ A ⋁ B = A))
43	42	orbi2i 255	. . 3 ⊢ (((A ∈ B ⋁ A = B) ⋁ (B ∈ A ⋁ A = B)) ↔ ((A ∈ B ⋁ A = B) ⋁ (B ∈ A ⋁ B = A)))
44	39, 40, 43	3bitr 177	. 2 ⊢ ((A ∈ B ⋁ A = B ⋁ B ∈ A) ↔ ((A ∈ B ⋁ A = B) ⋁ (B ∈ A ⋁ B = A)))
45	36, 44	sylibr 200	1 ⊢ ((Ord A ⋀ Ord B) → (A ∈ B ⋁ A = B ⋁ B ∈ A))

Syntax hints:  ¬ wn 2   → wi 3   ↔ wb 146   ⋁ wo 222   ⋀ wa 223   ⋁ w3o 774   = wceq 956   ∈ wcel 958   ∩ cin 2046   ⊆ wss 2047  Ord word 2947

This theorem is referenced by:  ordtri1 2980  ordtri3 2983  ordon 2987  ordeleqon 2990  ordtri2or 3077  zorn2lem6 4793

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-11 967  ax-12 968  ax-13 969  ax-14 970  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459  ax-sep 2703  ax-pow 2742  ax-pr 2779

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-3or 776  df-3an 777  df-ex 981  df-sb 1172  df-eu 1382  df-mo 1383  df-clab 1464  df-cleq 1469  df-clel 1472  df-ne 1587  df-ral 1649  df-rex 1650  df-v 1812  df-dif 2049  df-un 2050  df-in 2051  df-ss 2053  df-nul 2281  df-pw 2402  df-sn 2412  df-pr 2413  df-op 2416  df-uni 2504  df-br 2620  df-opab 2667  df-tr 2681  df-eprel 2832  df-po 2840  df-so 2850  df-fr 2917  df-we 2934  df-ord 2951

Copyright terms: Public domain