Theorem sotrieq 2149

Description: Trichotomy law for strict order relation.

Assertion

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

Proof of Theorem sotrieq

Step	Hyp	Ref	Expression
1		breq2 2066	. . . . . . . 8 ⊢ (B = C → (BRB ↔ BRC))
2	1	negbid 463	. . . . . . 7 ⊢ (B = C → (¬ BRB ↔ ¬ BRC))
3		sonr 2143	. . . . . . 7 ⊢ ((R Or A ∧ B ∈ A) → ¬ BRB)
4	2, 3	syl5bi 183	. . . . . 6 ⊢ (B = C → ((R Or A ∧ B ∈ A) → ¬ BRC))
5		breq2 2066	. . . . . . . 8 ⊢ (B = C → (CRB ↔ CRC))
6	5	negbid 463	. . . . . . 7 ⊢ (B = C → (¬ CRB ↔ ¬ CRC))
7		sonr 2143	. . . . . . 7 ⊢ ((R Or A ∧ C ∈ A) → ¬ CRC)
8	6, 7	syl5bir 184	. . . . . 6 ⊢ (B = C → ((R Or A ∧ C ∈ A) → ¬ CRB))
9	4, 8	anim12d 431	. . . . 5 ⊢ (B = C → (((R Or A ∧ B ∈ A) ∧ (R Or A ∧ C ∈ A)) → (¬ BRC ∧ ¬ CRB)))
10	9	com12 13	. . . 4 ⊢ (((R Or A ∧ B ∈ A) ∧ (R Or A ∧ C ∈ A)) → (B = C → (¬ BRC ∧ ¬ CRB)))
11	10	anandis 394	. . 3 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → (B = C → (¬ BRC ∧ ¬ CRB)))
12		sotric 2148	. . . . . . . . 9 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → (BRC ↔ ¬ (B = C ∨ CRB)))
13	12	bicon2d 404	. . . . . . . 8 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → ((B = C ∨ CRB) ↔ ¬ BRC))
14	13	biimpar 325	. . . . . . 7 ⊢ (((R Or A ∧ (B ∈ A ∧ C ∈ A)) ∧ ¬ BRC) → (B = C ∨ CRB))
15	14	ord 202	. . . . . 6 ⊢ (((R Or A ∧ (B ∈ A ∧ C ∈ A)) ∧ ¬ BRC) → (¬ B = C → CRB))
16	15	con1d 85	. . . . 5 ⊢ (((R Or A ∧ (B ∈ A ∧ C ∈ A)) ∧ ¬ BRC) → (¬ CRB → B = C))
17	16	exp 291	. . . 4 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → (¬ BRC → (¬ CRB → B = C)))
18	17	imp3a 279	. . 3 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → ((¬ BRC ∧ ¬ CRB) → B = C))
19	11, 18	impbid 397	. 2 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → (B = C ↔ (¬ BRC ∧ ¬ CRB)))
20		ioran 254	. 2 ⊢ (¬ (BRC ∨ CRB) ↔ (¬ BRC ∧ ¬ CRB))
21	19, 20	syl6bbr 416	1 ⊢ ((R Or A ∧ (B ∈ A ∧ C ∈ A)) → (B = C ↔ ¬ (BRC ∨ CRB)))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127   ∨ wo 195   ∧ wa 196   = wceq 1091   ∈ wcel 1092   class class class wbr 2054   Or wor 2059

This theorem is referenced by:  sotrieq2 2150  distrlem4pr 3924  addcanpr 3946  sqgt0sr 4009  lttri2t 4280

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-3an 583  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-po 2128  df-so 2138

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