Theorem issoi 5610

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 487	. . . 4 ⊢ ((⊤ ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
3		issoi.2	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
4	3	adantl 487	. . . 4 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
5	2, 4	ispod 5583	. . 3 ⊢ (⊤ → 𝑅 Po 𝐴)
6		issoi.3	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
7	6	adantl 487	. . 3 ⊢ ((⊤ ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
8	5, 7	issod 5609	. 2 ⊢ (⊤ → 𝑅 Or 𝐴)
9	8	mptru 1577	1 ⊢ 𝑅 Or 𝐴

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 401   ∨ w3o 1102   ∧ w3a 1103  ⊤wtru 1571   ∈ wcel 2146   class class class wbr 5114   Or wor 5573

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943

This proof depends on definitions:  df-bi 210  df-an 402  df-3an 1105  df-tru 1573  df-ral 3083  df-po 5574  df-so 5575

This theorem is used by:  isso2i  5611  ltsopr  11035  ltssolem1  27876