Theorem issod 4462

Description: An irreflexive, transitive, trichotomous relation is a linear ordering (in the sense of df-iso 4440). (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 1048	. . . . . . . . . 10 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥))
4		orc 724	. . . . . . . . . . . 12 ⊢ (𝑥𝑅𝑦 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))
5	4	a1i 9	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥𝑅𝑦 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
6		simp3r 1057	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑥𝑅𝑧)
7		breq1 4131	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝑧 ↔ 𝑦𝑅𝑧))
8	6, 7	syl5ibcom 155	. . . . . . . . . . . 12 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥 = 𝑦 → 𝑦𝑅𝑧))
9		olc 723	. . . . . . . . . . . 12 ⊢ (𝑦𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))
10	8, 9	syl6 33	. . . . . . . . . . 11 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑥 = 𝑦 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
11		simp1 1028	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝜑)
12		simp2r 1055	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑦 ∈ 𝐴)
13		simp2l 1054	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑥 ∈ 𝐴)
14		simp3l 1056	. . . . . . . . . . . . . 14 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → 𝑧 ∈ 𝐴)
15	12, 13, 14	3jca 1208	. . . . . . . . . . . . 13 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) ∧ (𝑧 ∈ 𝐴 ∧ 𝑥𝑅𝑧)) → (𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴))
16		potr 4451	. . . . . . . . . . . . . . . 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 1491	. . . . . . . . . . . . . 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 1277	. . . . . . . . . . . 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 1345	. . . . . . . . . 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 1234	. . . . . . . 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 2623	. . . . . 6 ⊢ ((𝜑 ∧ (𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴)) → ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
27	26	anassrs 404	. . . . 5 ⊢ (((𝜑 ∧ 𝑥 ∈ 𝐴) ∧ 𝑦 ∈ 𝐴) → ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
28	27	ralrimiva 2623	. . . 4 ⊢ ((𝜑 ∧ 𝑥 ∈ 𝐴) → ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
29		ralcom 2714	. . . 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 2623	. 2 ⊢ (𝜑 → ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧)))
32		df-iso 4440	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑧 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑧 → (𝑥𝑅𝑦 ∨ 𝑦𝑅𝑧))))
33	1, 31, 32	sylanbrc 421	1 ⊢ (𝜑 → 𝑅 Or 𝐴)

Syntax hints:   → wi 4   ∧ wa 104   ∨ wo 720   ∨ w3o 1008   ∧ w3a 1009   ∈ wcel 2209  ∀wral 2528   class class class wbr 4128   Po wpo 4437   Or wor 4438

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3or 1010  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ral 2533  df-v 2823  df-un 3224  df-sn 3714  df-pr 3715  df-op 3717  df-br 4129  df-po 4439  df-iso 4440

This theorem is referenced by:  ltsopi  7681  ltsonq  7759