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| Description: A strict order relation is a transitive relation. (Contributed by NM, 10-Feb-1996.) (Revised by Mario Carneiro, 10-May-2013.) | 
| Ref | Expression | 
|---|---|
| soi.1 | ⊢ 𝑅 Or 𝑆 | 
| soi.2 | ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | 
| Ref | Expression | 
|---|---|
| sotri | ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | soi.2 | . . . . 5 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
| 2 | 1 | brel 5749 | . . . 4 ⊢ (𝐴𝑅𝐵 → (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆)) | 
| 3 | 2 | simpld 494 | . . 3 ⊢ (𝐴𝑅𝐵 → 𝐴 ∈ 𝑆) | 
| 4 | 1 | brel 5749 | . . 3 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) | 
| 5 | 3, 4 | anim12i 613 | . 2 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → (𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆))) | 
| 6 | soi.1 | . . . 4 ⊢ 𝑅 Or 𝑆 | |
| 7 | sotr 5616 | . . . 4 ⊢ ((𝑅 Or 𝑆 ∧ (𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) | |
| 8 | 6, 7 | mpan 690 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ 𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) | 
| 9 | 8 | 3expb 1120 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) → ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶)) | 
| 10 | 5, 9 | mpcom 38 | 1 ⊢ ((𝐴𝑅𝐵 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) | 
| Colors of variables: wff setvar class | 
| Syntax hints: → wi 4 ∧ wa 395 ∧ w3a 1086 ∈ wcel 2107 ⊆ wss 3950 class class class wbr 5142 Or wor 5590 × cxp 5682 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 ax-sep 5295 ax-nul 5305 ax-pr 5431 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3an 1088 df-tru 1542 df-fal 1552 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-ral 3061 df-rex 3070 df-rab 3436 df-v 3481 df-dif 3953 df-un 3955 df-ss 3967 df-nul 4333 df-if 4525 df-sn 4626 df-pr 4628 df-op 4632 df-br 5143 df-opab 5205 df-po 5591 df-so 5592 df-xp 5690 | 
| This theorem is referenced by: son2lpi 6147 sotri2 6148 sotri3 6149 ltsonq 11010 ltbtwnnq 11019 nqpr 11055 prlem934 11074 ltexprlem4 11080 reclem2pr 11089 reclem4pr 11091 ltsosr 11135 addgt0sr 11145 supsrlem 11152 axpre-lttrn 11207 | 
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