Theorem swopo 5580

Description: A strict weak order is a partial order. (Contributed by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
swopo.1	⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
swopo.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))

Assertion

Ref	Expression
swopo	⊢ (𝜑 → 𝑅 Po 𝐴)

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

Proof of Theorem swopo

Step	Hyp	Ref	Expression
1		id 23	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
2	1	ancli 557	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3		swopo.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
4	3	ralrimivva 3208	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
5		breq1 5112	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑥𝑅𝑧))
6		breq2 5113	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑥))
7	6	notbid 321	. . . . . 6 ⊢ (𝑦 = 𝑥 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝑥))
8	5, 7	imbi12d 347	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦) ↔ (𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥)))
9		breq2 5113	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝑥))
10		breq1 5112	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑥 ↔ 𝑥𝑅𝑥))
11	10	notbid 321	. . . . . 6 ⊢ (𝑧 = 𝑥 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑥𝑅𝑥))
12	9, 11	imbi12d 347	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥) ↔ (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥)))
13	8, 12	rspc2va 3593	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
14	2, 4, 13	syl2anr 608	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
15	14	pm2.01d 192	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
16	3	3adantr1 1188	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
17		swopo.2	. . . . . . 7 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦)))
18	17	imp 411	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
19	18	orcomd 884	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑥𝑅𝑦) → (𝑧𝑅𝑦 ∨ 𝑥𝑅𝑧))
20	19	ord 877	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑥𝑅𝑦) → (¬ 𝑧𝑅𝑦 → 𝑥𝑅𝑧))
21	20	expimpd 458	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ ¬ 𝑧𝑅𝑦) → 𝑥𝑅𝑧))
22	16, 21	sylan2d 616	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
23	15, 22	ispod 5578	1 ⊢ (𝜑 → 𝑅 Po 𝐴)

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∧ wa 400   ∨ wo 860   ∧ w3a 1103   ∈ wcel 2143  ∀wral 3079   class class class wbr 5109   Po wpo 5567

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1825  ax-4 1839  ax-5 1940  ax-6 1997  ax-7 2038  ax-8 2145  ax-9 2153  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1810  df-sb 2097  df-clab 2742  df-cleq 2755  df-clel 2838  df-ral 3080  df-rab 3417  df-v 3457  df-dif 3908  df-un 3910  df-ss 3922  df-nul 4287  df-if 4488  df-sn 4590  df-pr 4592  df-op 4596  df-br 5110  df-po 5569

This theorem is used by:  swoer  8722  swoso  8725