Theorem soinxp 5758

Description: Intersection of total order with Cartesian 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 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		poinxp 5757	. . 3 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2		brinxp 5755	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3		biidd 262	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 ↔ 𝑥 = 𝑦))
4		brinxp 5755	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5	4	ancoms 460	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
6	2, 3, 5	3orbi123d 1436	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
7	6	ralbidva 3176	. . . 4 ⊢ (𝑥 ∈ 𝐴 → (∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
8	7	ralbiia 3092	. . 3 ⊢ (∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
9	1, 8	anbi12i 628	. 2 ⊢ ((𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)) ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
10		df-so 5590	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
11		df-so 5590	. 2 ⊢ ((𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴 ↔ ((𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
12	9, 10, 11	3bitr4i 303	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

Syntax hints:   ↔ wb 205   ∧ wa 397   ∨ w3o 1087   ∈ wcel 2107  ∀wral 3062   ∩ cin 3948   class class class wbr 5149   Po wpo 5587   Or wor 5588   × cxp 5675

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704  ax-sep 5300  ax-nul 5307  ax-pr 5428

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3or 1089  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-ral 3063  df-rex 3072  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-po 5589  df-so 5590  df-xp 5683

This theorem is referenced by:  weinxp  5761  ltsopi  10883  cnso  16190  opsrtoslem2  21617