Theorem solin 2145

Description: A strict order relation is linear (satisfies trichotomy).

Assertion

Ref	Expression
solin	⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → (BRC ∨ B = C ∨ CRB))

Proof of Theorem solin

Step	Hyp	Ref	Expression
1		breq1 2065	. . . . . 6 ⊢ (x = B → (xRy ↔ BRy))
2		cleq1 1107	. . . . . 6 ⊢ (x = B → (x = y ↔ B = y))
3		breq2 2066	. . . . . 6 ⊢ (x = B → (yRx ↔ yRB))
4	1, 2, 3	bi3ord 635	. . . . 5 ⊢ (x = B → ((xRy ∨ x = y ∨ yRx) ↔ (BRy ∨ B = y ∨ yRB)))
5	4	imbi2d 464	. . . 4 ⊢ (x = B → ((R Or A → (xRy ∨ x = y ∨ yRx)) ↔ (R Or A → (BRy ∨ B = y ∨ yRB))))
6		breq2 2066	. . . . . 6 ⊢ (y = C → (BRy ↔ BRC))
7		cleq2 1110	. . . . . 6 ⊢ (y = C → (B = y ↔ B = C))
8		breq1 2065	. . . . . 6 ⊢ (y = C → (yRB ↔ CRB))
9	6, 7, 8	bi3ord 635	. . . . 5 ⊢ (y = C → ((BRy ∨ B = y ∨ yRB) ↔ (BRC ∨ B = C ∨ CRB)))
10	9	imbi2d 464	. . . 4 ⊢ (y = C → ((R Or A → (BRy ∨ B = y ∨ yRB)) ↔ (R Or A → (BRC ∨ B = C ∨ CRB))))
11		df-so 2138	. . . . . 6 ⊢ (R Or A ↔ (R Po A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)))
12		ra42 1245	. . . . . . 7 ⊢ (∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx) → ((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx)))
13	12	adantl 305	. . . . . 6 ⊢ ((R Po A ∧ ∀x ∈ A ∀y ∈ A (xRy ∨ x = y ∨ yRx)) → ((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx)))
14	11, 13	sylbi 174	. . . . 5 ⊢ (R Or A → ((x ∈ A ∧ y ∈ A) → (xRy ∨ x = y ∨ yRx)))
15	14	com12 13	. . . 4 ⊢ ((x ∈ A ∧ y ∈ A) → (R Or A → (xRy ∨ x = y ∨ yRx)))
16	5, 10, 15	vtocl2ga 1388	. . 3 ⊢ ((B ∈ A ∧ C ∈ A) → (R Or A → (BRC ∨ B = C ∨ CRB)))
17	16	com12 13	. 2 ⊢ (R Or A → ((B ∈ A ∧ C ∈ A) → (BRC ∨ B = C ∨ CRB)))
18	17	imp 277	1 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → (BRC ∨ B = C ∨ CRB))

Syntax hints:   → wi 2   ∧ wa 196   ∨ w3o 580   = weq 797   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054   Po wpo 2058   Or wor 2059

This theorem is referenced by:  sotric 2148  dfwe2 2187  wecmpep 2193  wereu 2197

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-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3or 582  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-so 2138

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