Theorem isso2i 5596

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 2045	. . . . 5 ⊢ 𝑥 = 𝑥
2	1	orci 879	. . . 4 ⊢ (𝑥 = 𝑥 ∨ 𝑥𝑅𝑥)
3		nfv 1947	. . . . 5 ⊢ Ⅎ𝑦((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑥 ∨ 𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
4		eleq1w 2844	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (𝑦 ∈ 𝐴 ↔ 𝑥 ∈ 𝐴))
5	4	anbi2d 642	. . . . . 6 ⊢ (𝑦 = 𝑥 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴)))
6		equequ2 2059	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥 = 𝑦 ↔ 𝑥 = 𝑥))
7		breq1 5106	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑦𝑅𝑥 ↔ 𝑥𝑅𝑥))
8	6, 7	orbi12d 932	. . . . . . 7 ⊢ (𝑦 = 𝑥 → ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥 = 𝑥 ∨ 𝑥𝑅𝑥)))
9		breq2 5107	. . . . . . . 8 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥))
10	9	notbid 321	. . . . . . 7 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
11	8, 10	bibi12d 348	. . . . . 6 ⊢ (𝑦 = 𝑥 → (((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦) ↔ ((𝑥 = 𝑥 ∨ 𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥)))
12	5, 11	imbi12d 347	. . . . 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 357	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ¬ 𝑥𝑅𝑦))
15	3, 12, 14	chvarfv 2277	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ((𝑥 = 𝑥 ∨ 𝑥𝑅𝑥) ↔ ¬ 𝑥𝑅𝑥))
16	2, 15	mpbii 236	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
17	16	anidms 577	. 2 ⊢ (𝑥 ∈ 𝐴 → ¬ 𝑥𝑅𝑥)
18		isso2i.2	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))
19	14	biimprd 251	. . . 4 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (¬ 𝑥𝑅𝑦 → (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
20	19	orrd 877	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
21		3orass 1106	. . 3 ⊢ ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥𝑅𝑦 ∨ (𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
22	20, 21	sylibr 237	. 2 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
23	17, 18, 22	issoi 5595	1 ⊢ 𝑅 Or 𝐴

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 401   ∨ wo 861   ∨ w3o 1102   ∧ w3a 1103   ∈ wcel 2145   class class class wbr 5103   Or wor 5558

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  ax-6 2000  ax-7 2041  ax-8 2147  ax-9 2155  ax-12 2213  ax-ext 2733

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-nf 1817  df-sb 2100  df-clab 2740  df-cleq 2753  df-clel 2836  df-ral 3078  df-rab 3414  df-v 3453  df-dif 3902  df-un 3904  df-ss 3916  df-nul 4280  df-if 4483  df-sn 4585  df-pr 4587  df-op 4591  df-br 5104  df-po 5559  df-so 5560

This theorem is used by:  ltsonq  11047  ltsosr  11172  ltso  11383  xrltso  13263