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Theorem issoi 5218
Description: An irreflexive, transitive, linear relation is a strict ordering. (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)
Hypotheses
Ref Expression
issoi.1 (𝑥𝐴 → ¬ 𝑥𝑅𝑥)
issoi.2 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
issoi.3 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
Assertion
Ref Expression
issoi 𝑅 Or 𝐴
Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧

Proof of Theorem issoi
StepHypRef Expression
1 issoi.1 . . . . 5 (𝑥𝐴 → ¬ 𝑥𝑅𝑥)
21adantl 473 . . . 4 ((⊤ ∧ 𝑥𝐴) → ¬ 𝑥𝑅𝑥)
3 issoi.2 . . . . 5 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
43adantl 473 . . . 4 ((⊤ ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
52, 4ispod 5195 . . 3 (⊤ → 𝑅 Po 𝐴)
6 issoi.3 . . . 4 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
76adantl 473 . . 3 ((⊤ ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
85, 7issod 5217 . 2 (⊤ → 𝑅 Or 𝐴)
98trud 1642 1 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 383  w3o 1071  w3a 1072  wtru 1633  wcel 2139   class class class wbr 4804   Or wor 5186
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988
This theorem depends on definitions:  df-bi 197  df-an 385  df-3an 1074  df-tru 1635  df-ral 3055  df-po 5187  df-so 5188
This theorem is referenced by:  isso2i  5219  ltsopr  10066  sltsolem1  32153
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