Theorem pocl 2132

Description: Properties of partial order relation in class notation.

Assertion

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

Proof of Theorem pocl

Step	Hyp	Ref	Expression
1		id 9	. . . . . . 7 ⊢ (x = B → x = B)
2	1, 1	breq12d 2073	. . . . . 6 ⊢ (x = B → (xRx ↔ BRB))
3	2	negbid 463	. . . . 5 ⊢ (x = B → (¬ xRx ↔ ¬ BRB))
4		breq1 2065	. . . . . . 7 ⊢ (x = B → (xRy ↔ BRy))
5	4	anbi1d 469	. . . . . 6 ⊢ (x = B → ((xRy ∧ yRz) ↔ (BRy ∧ yRz)))
6		breq1 2065	. . . . . 6 ⊢ (x = B → (xRz ↔ BRz))
7	5, 6	imbi12d 474	. . . . 5 ⊢ (x = B → (((xRy ∧ yRz) → xRz) ↔ ((BRy ∧ yRz) → BRz)))
8	3, 7	anbi12d 476	. . . 4 ⊢ (x = B → ((¬ xRx ∧ ((xRy ∧ yRz) → xRz)) ↔ (¬ BRB ∧ ((BRy ∧ yRz) → BRz))))
9	8	imbi2d 464	. . 3 ⊢ (x = B → ((R Po A → (¬ xRx ∧ ((xRy ∧ yRz) → xRz))) ↔ (R Po A → (¬ BRB ∧ ((BRy ∧ yRz) → BRz)))))
10		breq2 2066	. . . . . . 7 ⊢ (y = C → (BRy ↔ BRC))
11		breq1 2065	. . . . . . 7 ⊢ (y = C → (yRz ↔ CRz))
12	10, 11	anbi12d 476	. . . . . 6 ⊢ (y = C → ((BRy ∧ yRz) ↔ (BRC ∧ CRz)))
13	12	imbi1d 465	. . . . 5 ⊢ (y = C → (((BRy ∧ yRz) → BRz) ↔ ((BRC ∧ CRz) → BRz)))
14	13	anbi2d 468	. . . 4 ⊢ (y = C → ((¬ BRB ∧ ((BRy ∧ yRz) → BRz)) ↔ (¬ BRB ∧ ((BRC ∧ CRz) → BRz))))
15	14	imbi2d 464	. . 3 ⊢ (y = C → ((R Po A → (¬ BRB ∧ ((BRy ∧ yRz) → BRz))) ↔ (R Po A → (¬ BRB ∧ ((BRC ∧ CRz) → BRz)))))
16		breq2 2066	. . . . . . 7 ⊢ (z = D → (CRz ↔ CRD))
17	16	anbi2d 468	. . . . . 6 ⊢ (z = D → ((BRC ∧ CRz) ↔ (BRC ∧ CRD)))
18		breq2 2066	. . . . . 6 ⊢ (z = D → (BRz ↔ BRD))
19	17, 18	imbi12d 474	. . . . 5 ⊢ (z = D → (((BRC ∧ CRz) → BRz) ↔ ((BRC ∧ CRD) → BRD)))
20	19	anbi2d 468	. . . 4 ⊢ (z = D → ((¬ BRB ∧ ((BRC ∧ CRz) → BRz)) ↔ (¬ BRB ∧ ((BRC ∧ CRD) → BRD))))
21	20	imbi2d 464	. . 3 ⊢ (z = D → ((R Po A → (¬ BRB ∧ ((BRC ∧ CRz) → BRz))) ↔ (R Po A → (¬ BRB ∧ ((BRC ∧ CRD) → BRD)))))
22		df-po 2128	. . . . . . . 8 ⊢ (R Po A ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (¬ xRx ∧ ((xRy ∧ yRz) → xRz)))
23		r3al 1240	. . . . . . . 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	bitr 151	. . . . . . 7 ⊢ (R Po A ↔ ∀x∀y∀z((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (¬ xRx ∧ ((xRy ∧ yRz) → xRz))))
25	24	biimp 133	. . . . . 6 ⊢ (R Po A → ∀x∀y∀z((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (¬ xRx ∧ ((xRy ∧ yRz) → xRz))))
26	25	19.21bbi 743	. . . . 5 ⊢ (R Po A → ∀z((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (¬ xRx ∧ ((xRy ∧ yRz) → xRz))))
27	26	19.21bi 742	. . . 4 ⊢ (R Po A → ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (¬ xRx ∧ ((xRy ∧ yRz) → xRz))))
28	27	com12 13	. . 3 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (R Po A → (¬ xRx ∧ ((xRy ∧ yRz) → xRz))))
29	9, 15, 21, 28	vtocl3ga 1389	. 2 ⊢ ((B ∈ A ∧ C ∈ A ∧ D ∈ A) → (R Po A → (¬ BRB ∧ ((BRC ∧ CRD) → BRD))))
30	29	com12 13	1 ⊢ (R Po A → ((B ∈ A ∧ C ∈ A ∧ D ∈ A) → (¬ BRB ∧ ((BRC ∧ CRD) → BRD))))

Syntax hints:  ¬ wn 1   → wi 2   ∧ wa 196   ∧ w3a 581  ∀wal 672   = wceq 1091   ∈ wcel 1092  ∀wral 1201   class class class wbr 2054   Po wpo 2058

This theorem is referenced by:  poirr 2133  potr 2134

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-3an 583  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-ral 1205  df-v 1349  df-un 1490  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-po 2128

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