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Mirrors > Home > MPE Home > Th. List > sotr | Structured version Visualization version GIF version |
Description: A strict order relation is a transitive relation. (Contributed by NM, 21-Jan-1996.) |
Ref | Expression |
---|---|
sotr | ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | sopo 5456 | . 2 ⊢ (𝑅 Or 𝐴 → 𝑅 Po 𝐴) | |
2 | potr 5450 | . 2 ⊢ ((𝑅 Po 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) | |
3 | 1, 2 | sylan 583 | 1 ⊢ ((𝑅 Or 𝐴 ∧ (𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴)) → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ∧ wa 399 ∧ w3a 1088 ∈ wcel 2113 class class class wbr 5027 Po wpo 5436 Or wor 5437 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1802 ax-4 1816 ax-5 1916 ax-6 1974 ax-7 2019 ax-8 2115 ax-9 2123 ax-10 2144 ax-11 2161 ax-12 2178 ax-ext 2710 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 847 df-3an 1090 df-tru 1545 df-ex 1787 df-nf 1791 df-sb 2074 df-clab 2717 df-cleq 2730 df-clel 2811 df-nfc 2881 df-ral 3058 df-v 3399 df-un 3846 df-sn 4514 df-pr 4516 df-op 4520 df-br 5028 df-po 5438 df-so 5439 |
This theorem is referenced by: sotr2 5469 wetrep 5512 wereu2 5516 sotri 5955 suplub2 8991 sotrd 33296 sotr3 33297 slttr 33583 fin2solem 35375 |
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