Theorem isso2i 5538

Description: Deduce strict ordering from its properties. (Contributed by NM, 29-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
isso2i.1	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
isso2i.2	⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))

Assertion

Ref	Expression
isso2i	⊢ 𝑅 Or 𝐴

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

Proof of Theorem isso2i

Step	Hyp	Ref	Expression
1		equid 2019	. . . . 5 ⊢ 𝑥 = 𝑥
2	1	orci 862	. . . 4 ⊢ (𝑥 = 𝑥 ∨ 𝑥𝑅𝑥)
3		nfv 1921	. . . . 5 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑥 ∨ 𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
4		eleq1w 2823	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5	4	anbi2d 629	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)))
6		equequ2 2033	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑥))
7		breq1 5082	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑥))
8	6, 7	orbi12d 916	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥 = 𝑥 ∨ 𝑥𝑅𝑥)))
9		breq2 5083	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥))
10	9	notbid 318	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
11	8, 10	bibi12d 346	. . . . . 6 ⊢ (𝑦 = 𝑥 → (((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦) ↔ ((𝑥 = 𝑥 ∨ 𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥)))
12	5, 11	imbi12d 345	. . . . 5 ⊢ (𝑦 = 𝑥 → (((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦)) ↔ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑥 ∨ 𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))))
13		isso2i.1	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ ¬ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
14	13	con2bid 355	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦))
15	3, 12, 14	chvarfv 2237	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑥 ∨ 𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
16	2, 15	mpbii 232	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
17	16	anidms 567	. 2 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥)
18		isso2i.2	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
19	14	biimprd 247	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥𝑅𝑦 → (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
20	19	orrd 860	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
21		3orass 1089	. . 3 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
22	20, 21	sylibr 233	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
23	17, 18, 22	issoi 5537	1 ⊢ 𝑅 Or 𝐴

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   ∨ wo 844   ∨ w3o 1085   ∧ w3a 1086   ∈ wcel 2110   class class class wbr 5079   Or wor 5502

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1975  ax-7 2015  ax-8 2112  ax-9 2120  ax-12 2175  ax-ext 2711

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3or 1087  df-3an 1088  df-tru 1545  df-fal 1555  df-ex 1787  df-nf 1791  df-sb 2072  df-clab 2718  df-cleq 2732  df-clel 2818  df-ral 3071  df-rab 3075  df-v 3433  df-dif 3895  df-un 3897  df-nul 4263  df-if 4466  df-sn 4568  df-pr 4570  df-op 4574  df-br 5080  df-po 5503  df-so 5504

This theorem is referenced by:  ltsonq  10724  ltsosr  10849  ltso  11054  xrltso  12872