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Theorem solin 5497
Description: A strict order relation is linear (satisfies trichotomy). (Contributed by NM, 21-Jan-1996.)
Assertion
Ref Expression
solin ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))

Proof of Theorem solin
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5066 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
2 eqeq1 2830 . . . . 5 (𝑥 = 𝐵 → (𝑥 = 𝑦𝐵 = 𝑦))
3 breq2 5067 . . . . 5 (𝑥 = 𝐵 → (𝑦𝑅𝑥𝑦𝑅𝐵))
41, 2, 33orbi123d 1428 . . . 4 (𝑥 = 𝐵 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵)))
54imbi2d 342 . . 3 (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵))))
6 breq2 5067 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
7 eqeq2 2838 . . . . 5 (𝑦 = 𝐶 → (𝐵 = 𝑦𝐵 = 𝐶))
8 breq1 5066 . . . . 5 (𝑦 = 𝐶 → (𝑦𝑅𝐵𝐶𝑅𝐵))
96, 7, 83orbi123d 1428 . . . 4 (𝑦 = 𝐶 → ((𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵) ↔ (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
109imbi2d 342 . . 3 (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))))
11 df-so 5474 . . . . 5 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
12 rsp2 3218 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
1311, 12simplbiim 505 . . . 4 (𝑅 Or 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
1413com12 32 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
155, 10, 14vtocl2ga 3580 . 2 ((𝐵𝐴𝐶𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
1615impcom 408 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3o 1080   = wceq 1530  wcel 2107  wral 3143   class class class wbr 5063   Po wpo 5471   Or wor 5472
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1789  ax-4 1803  ax-5 1904  ax-6 1963  ax-7 2008  ax-8 2109  ax-9 2117  ax-10 2138  ax-11 2153  ax-12 2169  ax-ext 2798
This theorem depends on definitions:  df-bi 208  df-an 397  df-or 844  df-3or 1082  df-3an 1083  df-tru 1533  df-ex 1774  df-nf 1778  df-sb 2063  df-clab 2805  df-cleq 2819  df-clel 2898  df-nfc 2968  df-ral 3148  df-rab 3152  df-v 3502  df-dif 3943  df-un 3945  df-in 3947  df-ss 3956  df-nul 4296  df-if 4471  df-sn 4565  df-pr 4567  df-op 4571  df-br 5064  df-so 5474
This theorem is referenced by:  sotric  5500  sotrieq  5501  somo  5509  wecmpep  5546  sorpssi  7445  soxp  7814  wfrlem10  7955  infsupprpr  8957  wemaplem2  9000  fpwwe2lem12  10052  fpwwe2lem13  10053  lttri4  10714  xmullem  12647  xmulasslem  12668  orngsqr  30791  noresle  33084  nosupbnd1lem6  33097  sltlin  33112  fin2so  34746  fnwe2lem3  39517  prproropf1olem4  43500
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