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Theorem solin 5570
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 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 breq1 5108 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
2 eqeq1 2740 . . . . 5 (𝑥 = 𝐵 → (𝑥 = 𝑦𝐵 = 𝑦))
3 breq2 5109 . . . . 5 (𝑥 = 𝐵 → (𝑦𝑅𝑥𝑦𝑅𝐵))
41, 2, 33orbi123d 1435 . . . 4 (𝑥 = 𝐵 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵)))
54imbi2d 340 . . 3 (𝑥 = 𝐵 → ((𝑅 Or 𝐴 → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵))))
6 breq2 5109 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
7 eqeq2 2748 . . . . 5 (𝑦 = 𝐶 → (𝐵 = 𝑦𝐵 = 𝐶))
8 breq1 5108 . . . . 5 (𝑦 = 𝐶 → (𝑦𝑅𝐵𝐶𝑅𝐵))
96, 7, 83orbi123d 1435 . . . 4 (𝑦 = 𝐶 → ((𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵) ↔ (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
109imbi2d 340 . . 3 (𝑦 = 𝐶 → ((𝑅 Or 𝐴 → (𝐵𝑅𝑦𝐵 = 𝑦𝑦𝑅𝐵)) ↔ (𝑅 Or 𝐴 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))))
11 df-so 5546 . . . . 5 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
12 breq1 5108 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
13 equequ1 2028 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
14 breq2 5109 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
1512, 13, 143orbi123d 1435 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑧𝑅𝑦𝑧 = 𝑦𝑦𝑅𝑧)))
1615ralbidv 3174 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑧𝑅𝑦𝑧 = 𝑦𝑦𝑅𝑧)))
1716rspw 3224 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝐴 → ∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
18 breq2 5109 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥𝑅𝑦𝑥𝑅𝑧))
19 equequ2 2029 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
20 breq1 5108 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
2118, 19, 203orbi123d 1435 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
2221rspw 3224 . . . . . . 7 (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑦𝐴 → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2317, 22syl6 35 . . . . . 6 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝐴 → (𝑦𝐴 → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))))
2423impd 411 . . . . 5 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2511, 24simplbiim 505 . . . 4 (𝑅 Or 𝐴 → ((𝑥𝐴𝑦𝐴) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
2625com12 32 . . 3 ((𝑥𝐴𝑦𝐴) → (𝑅 Or 𝐴 → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
275, 10, 26vtocl2ga 3535 . 2 ((𝐵𝐴𝐶𝐴) → (𝑅 Or 𝐴 → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵)))
2827impcom 408 1 ((𝑅 Or 𝐴 ∧ (𝐵𝐴𝐶𝐴)) → (𝐵𝑅𝐶𝐵 = 𝐶𝐶𝑅𝐵))
Colors of variables: wff setvar class
Syntax hints:  wi 4  wa 396  w3o 1086   = wceq 1541  wcel 2106  wral 3064   class class class wbr 5105   Po wpo 5543   Or wor 5544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2707
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-3or 1088  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2714  df-cleq 2728  df-clel 2814  df-ral 3065  df-rab 3408  df-v 3447  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4283  df-if 4487  df-sn 4587  df-pr 4589  df-op 4593  df-br 5106  df-so 5546
This theorem is referenced by:  sotric  5573  sotrieq  5574  somo  5582  wecmpep  5625  sorpssi  7666  soxp  8061  wfrlem10OLD  8264  infsupprpr  9440  wemaplem2  9483  fpwwe2lem11  10577  fpwwe2lem12  10578  lttri4  11239  xmullem  13183  xmulasslem  13204  noresle  27045  nosupbnd1lem6  27061  noinfbnd1lem6  27076  sltlin  27097  orngsqr  32099  fin2so  36065  fnwe2lem3  41365  prproropf1olem4  45688
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