Theorem pocl 4305

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 4018	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑥 ↔ 𝐵𝑅𝐵))
3	2	notbid 667	. . . . 5 ⊢ (𝑥 = 𝐵 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐵𝑅𝐵))
4		breq1 4008	. . . . . . 7 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑦 ↔ 𝐵𝑅𝑦))
5	4	anbi1d 465	. . . . . 6 ⊢ (𝑥 = 𝐵 → ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧)))
6		breq1 4008	. . . . . 6 ⊢ (𝑥 = 𝐵 → (𝑥𝑅𝑧 ↔ 𝐵𝑅𝑧))
7	5, 6	imbi12d 234	. . . . 5 ⊢ (𝑥 = 𝐵 → (((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)))
8	3, 7	anbi12d 473	. . . 4 ⊢ (𝑥 = 𝐵 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧))))
9	8	imbi2d 230	. . 3 ⊢ (𝑥 = 𝐵 → ((𝑅 Po 𝐴 → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)))))
10		breq2 4009	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝐵𝑅𝑦 ↔ 𝐵𝑅𝐶))
11		breq1 4008	. . . . . . 7 ⊢ (𝑦 = 𝐶 → (𝑦𝑅𝑧 ↔ 𝐶𝑅𝑧))
12	10, 11	anbi12d 473	. . . . . 6 ⊢ (𝑦 = 𝐶 → ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧)))
13	12	imbi1d 231	. . . . 5 ⊢ (𝑦 = 𝐶 → (((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)))
14	13	anbi2d 464	. . . 4 ⊢ (𝑦 = 𝐶 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧))))
15	14	imbi2d 230	. . 3 ⊢ (𝑦 = 𝐶 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)))))
16		breq2 4009	. . . . . . 7 ⊢ (𝑧 = 𝐷 → (𝐶𝑅𝑧 ↔ 𝐶𝑅𝐷))
17	16	anbi2d 464	. . . . . 6 ⊢ (𝑧 = 𝐷 → ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) ↔ (𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷)))
18		breq2 4009	. . . . . 6 ⊢ (𝑧 = 𝐷 → (𝐵𝑅𝑧 ↔ 𝐵𝑅𝐷))
19	17, 18	imbi12d 234	. . . . 5 ⊢ (𝑧 = 𝐷 → (((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧) ↔ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))
20	19	anbi2d 464	. . . 4 ⊢ (𝑧 = 𝐷 → ((¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧)) ↔ (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷))))
21	20	imbi2d 230	. . 3 ⊢ (𝑧 = 𝐷 → ((𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝑧) → 𝐵𝑅𝑧))) ↔ (𝑅 Po 𝐴 → (¬ 𝐵𝑅𝐵 ∧ ((𝐵𝑅𝐶 ∧ 𝐶𝑅𝐷) → 𝐵𝑅𝐷)))))
22		df-po 4298	. . . . . . . 8 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 ∀𝑧 ∈ 𝐴 (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
23		r3al 2521	. . . . . . . 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 184	. . . . . . 7 ⊢ (𝑅 Po 𝐴 ↔ ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
25	24	biimpi 120	. . . . . 6 ⊢ (𝑅 Po 𝐴 → ∀𝑥∀𝑦∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
26	25	19.21bbi 1559	. . . . 5 ⊢ (𝑅 Po 𝐴 → ∀𝑧((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴 ∧ 𝑧 ∈ 𝐴) → (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑧) → 𝑥𝑅𝑧))))
27	26	19.21bi 1558	. . . 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 2809	. 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 104   ∧ w3a 978  ∀wal 1351   = wceq 1353   ∈ wcel 2148  ∀wral 2455   class class class wbr 4005   Po wpo 4296

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2741  df-un 3135  df-sn 3600  df-pr 3601  df-op 3603  df-br 4006  df-po 4298

This theorem is referenced by:  poirr  4309  potr  4310