Theorem issod 4416

Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4394). (Contributed by NM, 21-Jan-1996.) (Revised by Mario Carneiro, 9-Jul-2014.)

Hypotheses

Ref	Expression
issod.1	⊢ (𝜑 → 𝑅 Po 𝐴)
issod.2	⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))

Assertion

Ref	Expression
issod	⊢ (𝜑 → 𝑅 Or 𝐴)

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

Proof of Theorem issod

Dummy variable 𝑧 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		issod.1	. 2 ⊢ (𝜑 → 𝑅 Po 𝐴)
2		issod.2	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
3	2	3adant3 1043	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
4		orc 719	. . . . . . . . . . . 12 ⊢ (𝑥𝑅𝑦 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))
5	4	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
6		simp3r 1052	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑥𝑅𝑧)
7		breq1 4091	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝑧 ↔ 𝑦𝑅𝑧))
8	6, 7	syl5ibcom 155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥 = 𝑦 → 𝑦𝑅𝑧))
9		olc 718	. . . . . . . . . . . 12 ⊢ (𝑦𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))
10	8, 9	syl6 33	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥 = 𝑦 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
11		simp1 1023	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝜑)
12		simp2r 1050	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑦 ∈ 𝐴)
13		simp2l 1049	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑥 ∈ 𝐴)
14		simp3l 1051	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑧 ∈ 𝐴)
15	12, 13, 14	3jca 1203	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
16		potr 4405	. . . . . . . . . . . . . . . 16 ⊢ ((𝑅 Po 𝐴 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑦𝑅𝑥 ∧ 𝑥𝑅𝑧) → 𝑦𝑅𝑧))
17	1, 16	sylan 283	. . . . . . . . . . . . . . 15 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → ((𝑦𝑅𝑥 ∧ 𝑥𝑅𝑧) → 𝑦𝑅𝑧))
18	17	expcomd 1486	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) → (𝑥𝑅𝑧 → (𝑦𝑅𝑥 → 𝑦𝑅𝑧)))
19	18	imp 124	. . . . . . . . . . . . 13 ⊢ (((𝜑 ∧ (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴)) ∧ 𝑥𝑅𝑧) → (𝑦𝑅𝑥 → 𝑦𝑅𝑧))
20	11, 15, 6, 19	syl21anc 1272	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑦𝑅𝑥 → 𝑦𝑅𝑧))
21	20, 9	syl6 33	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑦𝑅𝑥 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
22	5, 10, 21	3jaod 1340	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
23	3, 22	mpd 13	. . . . . . . . 9 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))
24	23	3expa 1229	. . . . . . . 8 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))
25	24	expr 375	. . . . . . 7 ⊢ (((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) ∧ 𝑧 ∈ 𝐴) → (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
26	25	ralrimiva 2605	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
27	26	anassrs 400	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
28	27	ralrimiva 2605	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
29		ralcom 2696	. . . 4 ⊢ (∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)) ↔ ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
30	28, 29	sylib 122	. . 3 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
31	30	ralrimiva 2605	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
32		df-iso 4394	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))))
33	1, 31, 32	sylanbrc 417	1 ⊢ (𝜑 → 𝑅 Or 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 715   ∨ w3o 1003   ∧ w3a 1004   ∈ wcel 2202  ∀wral 2510   class class class wbr 4088   Po wpo 4391   Or wor 4392

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3or 1005  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-ral 2515  df-v 2804  df-un 3204  df-sn 3675  df-pr 3676  df-op 3678  df-br 4089  df-po 4393  df-iso 4394

This theorem is referenced by:  ltsopi  7540  ltsonq  7618