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Theorem sosn 5775
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 4648 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2 elsni 4648 . . . . . . 7 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
32eqcomd 2741 . . . . . 6 (𝑦 ∈ {𝐴} → 𝐴 = 𝑦)
41, 3sylan9eq 2795 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
543mix2d 1336 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
65rgen2 3197 . . 3 𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
7 df-so 5598 . . 3 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
86, 7mpbiran2 710 . 2 (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴})
9 posn 5774 . 2 (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
108, 9bitrid 283 1 (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 206  wa 395  w3o 1085  wcel 2106  wral 3059  {csn 4631   class class class wbr 5148   Po wpo 5595   Or wor 5596  Rel wrel 5694
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706  ax-sep 5302  ax-nul 5312  ax-pr 5438
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-ral 3060  df-rex 3069  df-rab 3434  df-v 3480  df-dif 3966  df-un 3968  df-ss 3980  df-nul 4340  df-if 4532  df-sn 4632  df-pr 4634  df-op 4638  df-br 5149  df-opab 5211  df-po 5597  df-so 5598  df-xp 5695  df-rel 5696
This theorem is referenced by:  wesn  5777
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