Theorem swopo 4969

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 22	. . . . 5 ⊢ (𝑥 ∈ 𝐴 → 𝑥 ∈ 𝐴)
2	1	ancli 572	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴))
3		swopo.1	. . . . 5 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
4	3	ralrimivva 2954	. . . 4 ⊢ (𝜑 → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦))
5		breq1 4586	. . . . . 6 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑧 ↔ 𝑥𝑅𝑧))
6		breq2 4587	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑧𝑅𝑦 ↔ 𝑧𝑅𝑥))
7	6	notbid 307	. . . . . 6 ⊢ (𝑦 = 𝑥 → (¬ 𝑧𝑅𝑦 ↔ ¬ 𝑧𝑅𝑥))
8	5, 7	imbi12d 333	. . . . 5 ⊢ (𝑦 = 𝑥 → ((𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦) ↔ (𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥)))
9		breq2 4587	. . . . . 6 ⊢ (𝑧 = 𝑥 → (𝑥𝑅𝑧 ↔ 𝑥𝑅𝑥))
10		breq1 4586	. . . . . . 7 ⊢ (𝑧 = 𝑥 → (𝑧𝑅𝑥 ↔ 𝑥𝑅𝑥))
11	10	notbid 307	. . . . . 6 ⊢ (𝑧 = 𝑥 → (¬ 𝑧𝑅𝑥 ↔ ¬ 𝑥𝑅𝑥))
12	9, 11	imbi12d 333	. . . . 5 ⊢ (𝑧 = 𝑥 → ((𝑥𝑅𝑧 → ¬ 𝑧𝑅𝑥) ↔ (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥)))
13	8, 12	rspc2va 3294	. . . 4 ⊢ (((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) ∧ ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑦𝑅𝑧 → ¬ 𝑧𝑅𝑦)) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
14	2, 4, 13	syl2anr 494	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → (𝑥𝑅𝑥 → ¬ 𝑥𝑅𝑥))
15	14	pm2.01d 180	. 2 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
16	3	3adantr1 1213	. . 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 444	. . . . . 6 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑥𝑅𝑦) → (𝑥𝑅𝑧 ∨ 𝑧𝑅𝑦))
19	18	orcomd 402	. . . . 5 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑥𝑅𝑦) → (𝑧𝑅𝑦 ∨ 𝑥𝑅𝑧))
20	19	ord 391	. . . 4 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑥𝑅𝑦) → (¬ 𝑧𝑅𝑦 → 𝑥𝑅𝑧))
21	20	expimpd 627	. . 3 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ ¬ 𝑧𝑅𝑦) → 𝑥𝑅𝑧))
22	16, 21	sylan2d 498	. 2 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
23	15, 22	ispod 4967	1 ⊢ (𝜑 → 𝑅 Po 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∨ wo 382   ∧ wa 383   ∧ w3a 1031   ∈ wcel 1977  ∀wral 2896   class class class wbr 4583   Po wpo 4957

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  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-3an 1033  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-ral 2901  df-rab 2905  df-v 3175  df-dif 3543  df-un 3545  df-in 3547  df-ss 3554  df-nul 3875  df-if 4037  df-sn 4126  df-pr 4128  df-op 4132  df-br 4584  df-po 4959

This theorem is referenced by:  swoer  7659  swoso  7662