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Theorem sosn 5753
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 4638 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2 elsni 4638 . . . . . . 7 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
32eqcomd 2730 . . . . . 6 (𝑦 ∈ {𝐴} → 𝐴 = 𝑦)
41, 3sylan9eq 2784 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
543mix2d 1334 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
65rgen2 3189 . . 3 𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
7 df-so 5580 . . 3 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
86, 7mpbiran2 707 . 2 (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴})
9 posn 5752 . 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 205  wa 395  w3o 1083  wcel 2098  wral 3053  {csn 4621   class class class wbr 5139   Po wpo 5577   Or wor 5578  Rel wrel 5672
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 1905  ax-6 1963  ax-7 2003  ax-8 2100  ax-9 2108  ax-ext 2695  ax-sep 5290  ax-nul 5297  ax-pr 5418
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 845  df-3or 1085  df-3an 1086  df-tru 1536  df-fal 1546  df-ex 1774  df-sb 2060  df-clab 2702  df-cleq 2716  df-clel 2802  df-ral 3054  df-rex 3063  df-rab 3425  df-v 3468  df-dif 3944  df-un 3946  df-in 3948  df-ss 3958  df-nul 4316  df-if 4522  df-sn 4622  df-pr 4624  df-op 4628  df-br 5140  df-opab 5202  df-po 5579  df-so 5580  df-xp 5673  df-rel 5674
This theorem is referenced by:  wesn  5755
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