Theorem soinxp 4353

Description: Intersection of linear order with cross product of its field. (Contributed by Mario Carneiro, 10-Jul-2014.)

Assertion

Ref	Expression
soinxp	⊢ (𝑅 Or A ↔ (𝑅 ∩ (A × A)) Or A)

Proof of Theorem soinxp

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

Step	Hyp	Ref	Expression
1		poinxp 4352	. . 3 ⊢ (𝑅 Po A ↔ (𝑅 ∩ (A × A)) Po A)
2		brinxp 4351	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ A) → (x𝑅y ↔ x(𝑅 ∩ (A × A))y))
3	2	3adant3 923	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅y ↔ x(𝑅 ∩ (A × A))y))
4		brinxp 4351	. . . . . . . . 9 ⊢ ((x ∈ A ∧ z ∈ A) → (x𝑅z ↔ x(𝑅 ∩ (A × A))z))
5	4	3adant2 922	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (x𝑅z ↔ x(𝑅 ∩ (A × A))z))
6		brinxp 4351	. . . . . . . . . 10 ⊢ ((z ∈ A ∧ y ∈ A) → (z𝑅y ↔ z(𝑅 ∩ (A × A))y))
7	6	ancoms 255	. . . . . . . . 9 ⊢ ((y ∈ A ∧ z ∈ A) → (z𝑅y ↔ z(𝑅 ∩ (A × A))y))
8	7	3adant1 921	. . . . . . . 8 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → (z𝑅y ↔ z(𝑅 ∩ (A × A))y))
9	5, 8	orbi12d 706	. . . . . . 7 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → ((x𝑅z ∨ z𝑅y) ↔ (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y)))
10	3, 9	imbi12d 223	. . . . . 6 ⊢ ((x ∈ A ∧ y ∈ A ∧ z ∈ A) → ((x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ (x(𝑅 ∩ (A × A))y → (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y))))
11	10	3expb 1104	. . . . 5 ⊢ ((x ∈ A ∧ (y ∈ A ∧ z ∈ A)) → ((x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ (x(𝑅 ∩ (A × A))y → (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y))))
12	11	2ralbidva 2340	. . . 4 ⊢ (x ∈ A → (∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ ∀y ∈ A ∀z ∈ A (x(𝑅 ∩ (A × A))y → (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y))))
13	12	ralbiia 2332	. . 3 ⊢ (∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y)) ↔ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x(𝑅 ∩ (A × A))y → (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y)))
14	1, 13	anbi12i 433	. 2 ⊢ ((𝑅 Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y))) ↔ ((𝑅 ∩ (A × A)) Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x(𝑅 ∩ (A × A))y → (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y))))
15		df-iso 4025	. 2 ⊢ (𝑅 Or A ↔ (𝑅 Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x𝑅y → (x𝑅z ∨ z𝑅y))))
16		df-iso 4025	. 2 ⊢ ((𝑅 ∩ (A × A)) Or A ↔ ((𝑅 ∩ (A × A)) Po A ∧ ∀x ∈ A ∀y ∈ A ∀z ∈ A (x(𝑅 ∩ (A × A))y → (x(𝑅 ∩ (A × A))z ∨ z(𝑅 ∩ (A × A))y))))
17	14, 15, 16	3bitr4i 201	1 ⊢ (𝑅 Or A ↔ (𝑅 ∩ (A × A)) Or A)

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98   ∨ wo 628   ∧ w3a 884   ∈ wcel 1390  ∀wral 2300   ∩ cin 2910   class class class wbr 3755   Po wpo 4022   Or wor 4023   × cxp 4286

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-po 4024  df-iso 4025  df-xp 4294

This theorem is referenced by: (None)