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Mirrors > Home > ILE Home > Th. List > sotri2 | GIF version |
Description: A transitivity relation. (Read ¬ B < A and B < C implies A < C .) (Contributed by Mario Carneiro, 10-May-2013.) |
Ref | Expression |
---|---|
soi.1 | ⊢ 𝑅 Or 𝑆 |
soi.2 | ⊢ 𝑅 ⊆ (𝑆 × 𝑆) |
Ref | Expression |
---|---|
sotri2 | ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | simp2 987 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → ¬ 𝐵𝑅𝐴) | |
2 | soi.2 | . . . . . . 7 ⊢ 𝑅 ⊆ (𝑆 × 𝑆) | |
3 | 2 | brel 4650 | . . . . . 6 ⊢ (𝐵𝑅𝐶 → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) |
4 | 3 | 3ad2ant3 1009 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆)) |
5 | simp1 986 | . . . . 5 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴 ∈ 𝑆) | |
6 | df-3an 969 | . . . . 5 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) ↔ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆) ∧ 𝐴 ∈ 𝑆)) | |
7 | 4, 5, 6 | sylanbrc 414 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) |
8 | simp3 988 | . . . 4 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐵𝑅𝐶) | |
9 | soi.1 | . . . . 5 ⊢ 𝑅 Or 𝑆 | |
10 | sowlin 4292 | . . . . 5 ⊢ ((𝑅 Or 𝑆 ∧ (𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆)) → (𝐵𝑅𝐶 → (𝐵𝑅𝐴 ∨ 𝐴𝑅𝐶))) | |
11 | 9, 10 | mpan 421 | . . . 4 ⊢ ((𝐵 ∈ 𝑆 ∧ 𝐶 ∈ 𝑆 ∧ 𝐴 ∈ 𝑆) → (𝐵𝑅𝐶 → (𝐵𝑅𝐴 ∨ 𝐴𝑅𝐶))) |
12 | 7, 8, 11 | sylc 62 | . . 3 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (𝐵𝑅𝐴 ∨ 𝐴𝑅𝐶)) |
13 | 12 | ord 714 | . 2 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → (¬ 𝐵𝑅𝐴 → 𝐴𝑅𝐶)) |
14 | 1, 13 | mpd 13 | 1 ⊢ ((𝐴 ∈ 𝑆 ∧ ¬ 𝐵𝑅𝐴 ∧ 𝐵𝑅𝐶) → 𝐴𝑅𝐶) |
Colors of variables: wff set class |
Syntax hints: ¬ wn 3 → wi 4 ∧ wa 103 ∨ wo 698 ∧ w3a 967 ∈ wcel 2135 ⊆ wss 3111 class class class wbr 3976 Or wor 4267 × cxp 4596 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in2 605 ax-io 699 ax-5 1434 ax-7 1435 ax-gen 1436 ax-ie1 1480 ax-ie2 1481 ax-8 1491 ax-10 1492 ax-11 1493 ax-i12 1494 ax-bndl 1496 ax-4 1497 ax-17 1513 ax-i9 1517 ax-ial 1521 ax-i5r 1522 ax-14 2138 ax-ext 2146 ax-sep 4094 ax-pow 4147 ax-pr 4181 |
This theorem depends on definitions: df-bi 116 df-3an 969 df-tru 1345 df-nf 1448 df-sb 1750 df-clab 2151 df-cleq 2157 df-clel 2160 df-nfc 2295 df-ral 2447 df-rex 2448 df-v 2723 df-un 3115 df-in 3117 df-ss 3124 df-pw 3555 df-sn 3576 df-pr 3577 df-op 3579 df-br 3977 df-opab 4038 df-iso 4269 df-xp 4604 |
This theorem is referenced by: (None) |
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