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Theorem sotri3 5445

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 5090	. . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆))
3	2	simprd 478	. . 3 ⊢ (𝐴𝑅𝐵 → 𝐵 ∈ 𝑆)
4		soi.1	. . . . . . 7 ⊢ 𝑅 Or 𝑆
5		sotric 4985	. . . . . . 7 ⊢ ((𝑅 Or 𝑆 ∧ (𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
6	4, 5	mpan 702	. . . . . 6 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (𝐶𝑅𝐵 ↔ ¬ (𝐶 = 𝐵 ∨ 𝐵𝑅𝐶)))
7	6	con2bid 343	. . . . 5 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) ↔ ¬ 𝐶𝑅𝐵))
8		breq2 4587	. . . . . . 7 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐶 ↔ 𝐴𝑅𝐵))
9	8	biimprd 237	. . . . . 6 ⊢ (𝐶 = 𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
10	4, 1	sotri 5442	. . . . . . 7 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)
11	10	expcom 450	. . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
12	9, 11	jaoi 393	. . . . 5 ⊢ ((𝐶 = 𝐵 ∨ 𝐵𝑅𝐶) → (𝐴𝑅𝐵 → 𝐴𝑅𝐶))
13	7, 12	syl6bir 243	. . . 4 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → (𝐴𝑅𝐵 → 𝐴𝑅𝐶)))
14	13	com3r 85	. . 3 ⊢ (𝐴𝑅𝐵 → ((𝐶 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆) → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶)))
15	3, 14	mpan2d 706	. 2 ⊢ (𝐴𝑅𝐵 → (𝐶 ∈ 𝑆 → (¬ 𝐶𝑅𝐵 → 𝐴𝑅𝐶)))
16	15	3imp21 1269	1 ⊢ ((𝐶 ∈ 𝑆 ∧ 𝐴𝑅𝐵 ∧ ¬ 𝐶𝑅𝐵) → 𝐴𝑅𝐶)

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∧ w3a 1031 = wceq 1475 ∈ wcel 1977 ⊆ wss 3540 class class class wbr 4583 Or wor 4958 × cxp 5036

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pr 4833

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-sn 4126 df-pr 4128 df-op 4132 df-br 4584 df-opab 4644 df-po 4959 df-so 4960 df-xp 5044

This theorem is referenced by: archnq 9681

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