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Theorem issoi 5500
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 482 . . . 4 ((⊤ ∧ 𝑥𝐴) → ¬ 𝑥𝑅𝑥)
3 issoi.2 . . . . 5 ((𝑥𝐴𝑦𝐴𝑧𝐴) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
43adantl 482 . . . 4 ((⊤ ∧ (𝑥𝐴𝑦𝐴𝑧𝐴)) → ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧))
52, 4ispod 5475 . . 3 (⊤ → 𝑅 Po 𝐴)
6 issoi.3 . . . 4 ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
76adantl 482 . . 3 ((⊤ ∧ (𝑥𝐴𝑦𝐴)) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
85, 7issod 5499 . 2 (⊤ → 𝑅 Or 𝐴)
98mptru 1535 1 𝑅 Or 𝐴
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wa 396  w3o 1078  w3a 1079  wtru 1529  wcel 2105   class class class wbr 5057   Or wor 5466
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1787  ax-4 1801  ax-5 1902
This theorem depends on definitions:  df-bi 208  df-an 397  df-3an 1081  df-tru 1531  df-ral 3140  df-po 5467  df-so 5468
This theorem is referenced by:  isso2i  5501  ltsopr  10442  sltsolem1  33077
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