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Mirrors > Home > MPE Home > Th. List > Mathboxes > sotrd | Structured version Visualization version GIF version |
Description: Transitivity law for strict orderings, deduction form. (Contributed by Scott Fenton, 24-Nov-2021.) |
Ref | Expression |
---|---|
sotrd.1 | ⊢ (𝜑 → 𝑅 Or 𝐴) |
sotrd.2 | ⊢ (𝜑 → 𝑋 ∈ 𝐴) |
sotrd.3 | ⊢ (𝜑 → 𝑌 ∈ 𝐴) |
sotrd.4 | ⊢ (𝜑 → 𝑍 ∈ 𝐴) |
sotrd.5 | ⊢ (𝜑 → 𝑋𝑅𝑌) |
sotrd.6 | ⊢ (𝜑 → 𝑌𝑅𝑍) |
Ref | Expression |
---|---|
sotrd | ⊢ (𝜑 → 𝑋𝑅𝑍) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sotrd.5 | . 2 ⊢ (𝜑 → 𝑋𝑅𝑌) | |
2 | sotrd.6 | . 2 ⊢ (𝜑 → 𝑌𝑅𝑍) | |
3 | sotrd.1 | . . 3 ⊢ (𝜑 → 𝑅 Or 𝐴) | |
4 | sotrd.2 | . . 3 ⊢ (𝜑 → 𝑋 ∈ 𝐴) | |
5 | sotrd.3 | . . 3 ⊢ (𝜑 → 𝑌 ∈ 𝐴) | |
6 | sotrd.4 | . . 3 ⊢ (𝜑 → 𝑍 ∈ 𝐴) | |
7 | sotr 5603 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍)) | |
8 | 3, 4, 5, 6, 7 | syl13anc 1369 | . 2 ⊢ (𝜑 → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍)) |
9 | 1, 2, 8 | mp2and 696 | 1 ⊢ (𝜑 → 𝑋𝑅𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 395 ∈ wcel 2098 class class class wbr 5139 Or wor 5578 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1789 ax-4 1803 ax-5 1905 ax-6 1963 ax-7 2003 ax-8 2100 ax-9 2108 ax-ext 2695 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 845 df-3an 1086 df-tru 1536 df-fal 1546 df-ex 1774 df-sb 2060 df-clab 2702 df-cleq 2716 df-clel 2802 df-ral 3054 df-rab 3425 df-v 3468 df-dif 3944 df-un 3946 df-in 3948 df-ss 3958 df-nul 4316 df-if 4522 df-sn 4622 df-pr 4624 df-op 4628 df-br 5140 df-po 5579 df-so 5580 |
This theorem is referenced by: (None) |
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