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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 5057 | . . 3 ⊢ ((𝑅 Or 𝐴 ∧ (𝑋 ∈ 𝐴 ∧ 𝑌 ∈ 𝐴 ∧ 𝑍 ∈ 𝐴)) → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍)) | |
8 | 3, 4, 5, 6, 7 | syl13anc 1328 | . 2 ⊢ (𝜑 → ((𝑋𝑅𝑌 ∧ 𝑌𝑅𝑍) → 𝑋𝑅𝑍)) |
9 | 1, 2, 8 | mp2and 715 | 1 ⊢ (𝜑 → 𝑋𝑅𝑍) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 384 ∈ wcel 1990 class class class wbr 4653 Or wor 5034 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-po 5035 df-so 5036 |
This theorem is referenced by: (None) |
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