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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.
StepHypRef Expression
1 poinxp 4352 . . 3 (𝑅 Po A ↔ (𝑅 ∩ (A × A)) Po A)
2 brinxp 4351 . . . . . . . 8 ((x A y A) → (x𝑅yx(𝑅 ∩ (A × A))y))
323adant3 923 . . . . . . 7 ((x A y A z A) → (x𝑅yx(𝑅 ∩ (A × A))y))
4 brinxp 4351 . . . . . . . . 9 ((x A z A) → (x𝑅zx(𝑅 ∩ (A × A))z))
543adant2 922 . . . . . . . 8 ((x A y A z A) → (x𝑅zx(𝑅 ∩ (A × A))z))
6 brinxp 4351 . . . . . . . . . 10 ((z A y A) → (z𝑅yz(𝑅 ∩ (A × A))y))
76ancoms 255 . . . . . . . . 9 ((y A z A) → (z𝑅yz(𝑅 ∩ (A × A))y))
873adant1 921 . . . . . . . 8 ((x A y A z A) → (z𝑅yz(𝑅 ∩ (A × A))y))
95, 8orbi12d 706 . . . . . . 7 ((x A y A z A) → ((x𝑅z z𝑅y) ↔ (x(𝑅 ∩ (A × A))z z(𝑅 ∩ (A × A))y)))
103, 9imbi12d 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))))
11103expb 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))))
12112ralbidva 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))))
1312ralbiia 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)))
141, 13anbi12i 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))))
1714, 15, 163bitr4i 201 1 (𝑅 Or A ↔ (𝑅 ∩ (A × A)) Or A)
Colors of variables: wff set class
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-bnd 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)
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