MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  solin Structured version   Visualization version   GIF version

Theorem solin 5612
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 5150 . . . . 5 (𝑥 = 𝐵 → (𝑥𝑅𝑦𝐵𝑅𝑦))
2 eqeq1 2736 . . . . 5 (𝑥 = 𝐵 → (𝑥 = 𝑦𝐵 = 𝑦))
3 breq2 5151 . . . . 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 5151 . . . . 5 (𝑦 = 𝐶 → (𝐵𝑅𝑦𝐵𝑅𝐶))
7 eqeq2 2744 . . . . 5 (𝑦 = 𝐶 → (𝐵 = 𝑦𝐵 = 𝐶))
8 breq1 5150 . . . . 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 5588 . . . . 5 (𝑅 Or 𝐴 ↔ (𝑅 Po 𝐴 ∧ ∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
12 breq1 5150 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥𝑅𝑦𝑧𝑅𝑦))
13 equequ1 2028 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑥 = 𝑦𝑧 = 𝑦))
14 breq2 5151 . . . . . . . . . 10 (𝑥 = 𝑧 → (𝑦𝑅𝑥𝑦𝑅𝑧))
1512, 13, 143orbi123d 1435 . . . . . . . . 9 (𝑥 = 𝑧 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑧𝑅𝑦𝑧 = 𝑦𝑦𝑅𝑧)))
1615ralbidv 3177 . . . . . . . 8 (𝑥 = 𝑧 → (∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ ∀𝑦𝐴 (𝑧𝑅𝑦𝑧 = 𝑦𝑦𝑅𝑧)))
1716rspw 3233 . . . . . . 7 (∀𝑥𝐴𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) → (𝑥𝐴 → ∀𝑦𝐴 (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
18 breq2 5151 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥𝑅𝑦𝑥𝑅𝑧))
19 equequ2 2029 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑥 = 𝑦𝑥 = 𝑧))
20 breq1 5150 . . . . . . . . 9 (𝑦 = 𝑧 → (𝑦𝑅𝑥𝑧𝑅𝑥))
2118, 19, 203orbi123d 1435 . . . . . . . 8 (𝑦 = 𝑧 → ((𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥) ↔ (𝑥𝑅𝑧𝑥 = 𝑧𝑧𝑅𝑥)))
2221rspw 3233 . . . . . . 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 3566 . 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 3061   class class class wbr 5147   Po wpo 5585   Or wor 5586
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 2703
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 2710  df-cleq 2724  df-clel 2810  df-ral 3062  df-rab 3433  df-v 3476  df-dif 3950  df-un 3952  df-in 3954  df-ss 3964  df-nul 4322  df-if 4528  df-sn 4628  df-pr 4630  df-op 4634  df-br 5148  df-so 5588
This theorem is referenced by:  sotric  5615  sotrieq  5616  somo  5624  wecmpep  5667  sorpssi  7715  soxp  8111  wfrlem10OLD  8314  infsupprpr  9495  wemaplem2  9538  fpwwe2lem11  10632  fpwwe2lem12  10633  lttri4  11294  xmullem  13239  xmulasslem  13260  noresle  27189  nosupbnd1lem6  27205  noinfbnd1lem6  27220  sltlin  27241  orngsqr  32410  fin2so  36463  fnwe2lem3  41779  prproropf1olem4  46160
  Copyright terms: Public domain W3C validator