Theorem pocl 4225

Description: Properties of partial order relation in class notation. (Contributed by NM, 27-Mar-1997.)

Assertion

Ref	Expression
pocl	⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Proof of Theorem pocl

Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		id 19	. . . . . . 7 ⊢ (𝑥 = 𝐵 → 𝑥 = 𝐵)
2	1, 1	breq12d 3942	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑥 ↔ 𝐵𝑅𝐵))
3	2	notbid 656	. . . . 5 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
4		breq1 3932	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
5	4	anbi1d 460	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧)))
6		breq1 3932	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑧 ↔ 𝐵𝑅𝑧))
7	5, 6	imbi12d 233	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)))
8	3, 7	anbi12d 464	. . . 4 ⊢ (𝑥 = 𝐵 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧))))
9	8	imbi2d 229	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Po 𝐴 → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)))))
10		breq2 3933	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
11		breq1 3932	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝐶𝑅𝑧))
12	10, 11	anbi12d 464	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧)))
13	12	imbi1d 230	. . . . 5 ⊢ (𝑦 = 𝐶 → (((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)))
14	13	anbi2d 459	. . . 4 ⊢ (𝑦 = 𝐶 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧))))
15	14	imbi2d 229	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)))))
16		breq2 3933	. . . . . . 7 ⊢ (𝑧 = 𝐷 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝐷))
17	16	anbi2d 459	. . . . . 6 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷)))
18		breq2 3933	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝐵𝑅𝑧 ↔ 𝐵𝑅𝐷))
19	17, 18	imbi12d 233	. . . . 5 ⊢ (𝑧 = 𝐷 → (((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))
20	19	anbi2d 459	. . . 4 ⊢ (𝑧 = 𝐷 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
21	20	imbi2d 229	. . 3 ⊢ (𝑧 = 𝐷 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))))
22		df-po 4218	. . . . . . . 8 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
23		r3al 2477	. . . . . . . 8 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
24	22, 23	bitri 183	. . . . . . 7 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
25	24	biimpi 119	. . . . . 6 ⊢ (𝑅 Po 𝐴 → ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
26	25	19.21bbi 1538	. . . . 5 ⊢ (𝑅 Po 𝐴 → ∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
27	26	19.21bi 1537	. . . 4 ⊢ (𝑅 Po 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
28	27	com12 30	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (𝑅 Po 𝐴 → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
29	9, 15, 21, 28	vtocl3ga 2756	. 2 ⊢ ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
30	29	com12 30	1 ⊢ (𝑅 Po 𝐴 → ((𝐵 ∈ 𝐴 ∧ 𝐶 ∈ 𝐴 ∧ 𝐷 ∈ 𝐴) → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ∧ w3a 962  ∀wal 1329   = wceq 1331   ∈ wcel 1480  ∀wral 2416   class class class wbr 3929   Po wpo 4216

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218

This theorem is referenced by:  poirr  4229  potr  4230