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

Theorem sosn 5717
Description: Strict ordering on a singleton. (Contributed by Mario Carneiro, 28-Dec-2014.)
Assertion
Ref Expression
sosn (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem sosn
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 elsni 4602 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2 elsni 4602 . . . . . . 7 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
32eqcomd 2742 . . . . . 6 (𝑦 ∈ {𝐴} → 𝐴 = 𝑦)
41, 3sylan9eq 2796 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
543mix2d 1337 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
65rgen2 3193 . . 3 𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
7 df-so 5545 . . 3 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
86, 7mpbiran2 708 . 2 (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴})
9 posn 5716 . 2 (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
108, 9bitrid 282 1 (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396  w3o 1086  wcel 2106  wral 3063  {csn 4585   class class class wbr 5104   Po wpo 5542   Or wor 5543  Rel wrel 5637
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  ax-sep 5255  ax-nul 5262  ax-pr 5383
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 3064  df-rex 3073  df-rab 3407  df-v 3446  df-dif 3912  df-un 3914  df-in 3916  df-ss 3926  df-nul 4282  df-if 4486  df-sn 4586  df-pr 4588  df-op 4592  df-br 5105  df-opab 5167  df-po 5544  df-so 5545  df-xp 5638  df-rel 5639
This theorem is referenced by:  wesn  5719
  Copyright terms: Public domain W3C validator