Theorem sotri3 6087

Description: A transitivity relation. (Read 𝐴 < 𝐵 and 𝐵 ≤ 𝐶 implies 𝐴 < 𝐶.) (Contributed by Mario Carneiro, 10-May-2013.)

Hypotheses

Ref	Expression
soi.1	⊢ 𝑅 Or 𝑆
soi.2	⊢ 𝑅 ⊆ (𝑆 × 𝑆)

Assertion

Ref	Expression
sotri3	⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Proof of Theorem sotri3

Step	Hyp	Ref	Expression
1		soi.2	. . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆)
2	1	brel 5686	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3	2	simprd 497	. . 3 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ 𝑆)
4		soi.1	. . . . . . 7 ⊢ 𝑅 Or 𝑆
5		sotric 5559	. . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
6	4, 5	mpan 697	. . . . . 6 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
7	6	con2bid 356	. . . . 5 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8		breq2 5079	. . . . . . 7 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐴𝑅𝐵))
9	8	biimprd 250	. . . . . 6 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
10	4, 1	sotri 6084	. . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
11	10	expcom 415	. . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
12	9, 11	jaoi 864	. . . . 5 ⊢ ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
13	7, 12	biimtrrdi 256	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)))
14	13	com3r 87	. . 3 ⊢ (𝐴𝑅𝐵 → ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶)))
15	3, 14	mpan2d 701	. 2 ⊢ (𝐴𝑅𝐵 → (𝐶 ∈ 𝑆 → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶)))
16	15	3imp21 1120	1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 397   ∨ wo 854   ∧ w3a 1093   = wceq 1548   ∈ wcel 2121   ⊆ wss 3885   class class class wbr 5075   Or wor 5528   × cxp 5619

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5221  ax-pr 5365

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3or 1094  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3888  df-un 3890  df-in 3892  df-ss 3902  df-nul 4265  df-if 4458  df-sn 4559  df-pr 4561  df-op 4565  df-br 5076  df-opab 5138  df-po 5529  df-so 5530  df-xp 5627

This theorem is referenced by:  archnq  10898