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Theorem soinxp 4333
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 4332 . . 3 (𝑅 Po A ↔ (𝑅 ∩ (A × A)) Po A)
2 brinxp 4331 . . . . . . . 8 ((x A y A) → (x𝑅yx(𝑅 ∩ (A × A))y))
323adant3 910 . . . . . . 7 ((x A y A z A) → (x𝑅yx(𝑅 ∩ (A × A))y))
4 brinxp 4331 . . . . . . . . 9 ((x A z A) → (x𝑅zx(𝑅 ∩ (A × A))z))
543adant2 909 . . . . . . . 8 ((x A y A z A) → (x𝑅zx(𝑅 ∩ (A × A))z))
6 brinxp 4331 . . . . . . . . . 10 ((z A y A) → (z𝑅yz(𝑅 ∩ (A × A))y))
76ancoms 255 . . . . . . . . 9 ((y A z A) → (z𝑅yz(𝑅 ∩ (A × A))y))
873adant1 908 . . . . . . . 8 ((x A y A z A) → (z𝑅yz(𝑅 ∩ (A × A))y))
95, 8orbi12d 694 . . . . . . 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 1089 . . . . 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 2320 . . . 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 2312 . . 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 436 . 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 4004 . 2 (𝑅 Or A ↔ (𝑅 Po A x A y A z A (x𝑅y → (x𝑅z z𝑅y))))
16 df-iso 4004 . 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 616   w3a 871   wcel 1370  wral 2280  cin 2889   class class class wbr 3734   Po wpo 4001   Or wor 4002   × cxp 4266
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 532  ax-in2 533  ax-io 617  ax-5 1312  ax-7 1313  ax-gen 1314  ax-ie1 1359  ax-ie2 1360  ax-8 1372  ax-10 1373  ax-11 1374  ax-i12 1375  ax-bnd 1376  ax-4 1377  ax-14 1382  ax-17 1396  ax-i9 1400  ax-ial 1405  ax-i5r 1406  ax-ext 2000  ax-sep 3845  ax-pow 3897  ax-pr 3914
This theorem depends on definitions:  df-bi 110  df-3an 873  df-tru 1229  df-nf 1326  df-sb 1624  df-clab 2005  df-cleq 2011  df-clel 2014  df-nfc 2145  df-ral 2285  df-rex 2286  df-v 2533  df-un 2895  df-in 2897  df-ss 2904  df-pw 3332  df-sn 3352  df-pr 3353  df-op 3355  df-br 3735  df-opab 3789  df-po 4003  df-iso 4004  df-xp 4274
This theorem is referenced by: (None)
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