Theorem soinxp 5748

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 5747	. . 3 ⊢ (𝑅 Po 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Po 𝐴)
2		brinxp 5745	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥𝑅𝑦 ↔ 𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦))
3		biidd 265	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑥 = 𝑦 ↔ 𝑥 = 𝑦))
4		brinxp 5745	. . . . . . 7 ⊢ ((𝑦 ∈ 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
5	4	ancoms 464	. . . . . 6 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → (𝑦𝑅𝑥 ↔ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥))
6	2, 3, 5	3orbi123d 1463	. . . . 5 ⊢ ((𝑥 ∈ 𝐴 ∧ 𝑦 ∈ 𝐴) → ((𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥) ↔ (𝑥(𝑅 ∩ (𝐴 × 𝐴))𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦(𝑅 ∩ (𝐴 × 𝐴))𝑥)))
7	6	ralbidva 3189	. . . 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 3112	. . 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 640	. 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 5575	. 2 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥 ∈ 𝐴 ∀𝑦 ∈ 𝐴 (𝑥𝑅𝑦 ∨ 𝑥 = 𝑦 ∨ 𝑦𝑅𝑥)))
11		df-so 5575	. 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 306	1 ⊢ (𝑅 Or 𝐴 ↔ (𝑅 ∩ (𝐴 × 𝐴)) Or 𝐴)

This proof depends on syntax axioms:   ↔ wb 209   ∧ wa 401   ∨ w3o 1102   ∈ wcel 2146  ∀wral 3082   ∩ cin 3907   class class class wbr 5114   Po wpo 5572   Or wor 5573   × cxp 5664

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738  ax-sep 5262  ax-pr 5409

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-3or 1104  df-3an 1105  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-ral 3083  df-rex 3093  df-rab 3420  df-v 3460  df-dif 3911  df-un 3913  df-in 3915  df-ss 3925  df-nul 4290  df-if 4493  df-sn 4595  df-pr 4597  df-op 4601  df-br 5115  df-opab 5179  df-po 5574  df-so 5575  df-xp 5672

This theorem is used by:  weinxp  5751  ltsopi  10891  cnso  16328  opsrtoslem2  22244