Theorem sotricim 4051

Description: One direction of sotritric 4052 holds for all weakly linear orders. (Contributed by Jim Kingdon, 28-Sep-2019.)

Assertion

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

Proof of Theorem sotricim

Step	Hyp	Ref	Expression
1		sonr 4045	. . . . . . 7 ⊢ ((𝑅 Or A ∧ B ∈ A) → ¬ B𝑅B)
2	1	adantrr 448	. . . . . 6 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → ¬ B𝑅B)
3	2	3adant3 923	. . . . 5 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A) ∧ B𝑅𝐶) → ¬ B𝑅B)
4		breq2 3759	. . . . . . 7 ⊢ (B = 𝐶 → (B𝑅B ↔ B𝑅𝐶))
5	4	biimprcd 149	. . . . . 6 ⊢ (B𝑅𝐶 → (B = 𝐶 → B𝑅B))
6	5	3ad2ant3 926	. . . . 5 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A) ∧ B𝑅𝐶) → (B = 𝐶 → B𝑅B))
7	3, 6	mtod 588	. . . 4 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A) ∧ B𝑅𝐶) → ¬ B = 𝐶)
8	7	3expia 1105	. . 3 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → (B𝑅𝐶 → ¬ B = 𝐶))
9		so2nr 4049	. . . 4 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → ¬ (B𝑅𝐶 ∧ 𝐶𝑅B))
10		imnan 623	. . . 4 ⊢ ((B𝑅𝐶 → ¬ 𝐶𝑅B) ↔ ¬ (B𝑅𝐶 ∧ 𝐶𝑅B))
11	9, 10	sylibr 137	. . 3 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → (B𝑅𝐶 → ¬ 𝐶𝑅B))
12	8, 11	jcad 291	. 2 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → (B𝑅𝐶 → (¬ B = 𝐶 ∧ ¬ 𝐶𝑅B)))
13		ioran 668	. 2 ⊢ (¬ (B = 𝐶 ∨ 𝐶𝑅B) ↔ (¬ B = 𝐶 ∧ ¬ 𝐶𝑅B))
14	12, 13	syl6ibr 151	1 ⊢ ((𝑅 Or A ∧ (B ∈ A ∧ 𝐶 ∈ A)) → (B𝑅𝐶 → ¬ (B = 𝐶 ∨ 𝐶𝑅B)))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 97   ∨ wo 628   ∧ w3a 884   = wceq 1242   ∈ wcel 1390   class class class wbr 3755   Or wor 4023

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-v 2553  df-un 2916  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-po 4024  df-iso 4025

This theorem is referenced by:  sotritric  4052