Theorem issoi 4990

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	⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥)
issoi.2	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
issoi.3	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))

Assertion

Ref	Expression
issoi	⊢ 𝑅 Or 𝐴

Distinct variable groups:   𝑥,𝑦,𝑧,𝑅   𝑥,𝐴,𝑦,𝑧

Proof of Theorem issoi

Step	Hyp	Ref	Expression
1		issoi.1	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥)
2	1	adantl 481	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
3		issoi.2	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
4	3	adantl 481	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
5	2, 4	ispod 4967	. . 3 ⊢ (⊤ → 𝑅 Po 𝐴)
6		issoi.3	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
7	6	adantl 481	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
8	5, 7	issod 4989	. 2 ⊢ (⊤ → 𝑅 Or 𝐴)
9	8	trud 1484	1 ⊢ 𝑅 Or 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 383   ∨ w3o 1030   ∧ w3a 1031  ⊤wtru 1476   ∈ wcel 1977   class class class wbr 4583   Or wor 4958

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827

This theorem depends on definitions:  df-bi 196  df-an 385  df-3an 1033  df-tru 1478  df-ral 2901  df-po 4959  df-so 4960

This theorem is referenced by:  isso2i  4991  ltsopr  9733  sltsolem1  31067