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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 5497 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍)) | |
8 | 3, 4, 5, 6, 7 | syl13anc 1368 | . 2 ⊢ (𝜑 → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍)) |
9 | 1, 2, 8 | mp2and 697 | 1 ⊢ (𝜑 → 𝑋𝑅𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 398 ∈ wcel 2114 class class class wbr 5066 Or wor 5473 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1796 ax-4 1810 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2116 ax-9 2124 ax-10 2145 ax-11 2161 ax-12 2177 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-3an 1085 df-tru 1540 df-ex 1781 df-nf 1785 df-sb 2070 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-ral 3143 df-rab 3147 df-v 3496 df-dif 3939 df-un 3941 df-in 3943 df-ss 3952 df-nul 4292 df-if 4468 df-sn 4568 df-pr 4570 df-op 4574 df-br 5067 df-po 5474 df-so 5475 |
This theorem is referenced by: (None) |
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