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Theorem sosn 5758
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 4641 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2 elsni 4641 . . . . . . 7 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
32eqcomd 2734 . . . . . 6 (𝑦 ∈ {𝐴} → 𝐴 = 𝑦)
41, 3sylan9eq 2788 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
543mix2d 1335 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
65rgen2 3193 . . 3 𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
7 df-so 5585 . . 3 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
86, 7mpbiran2 709 . 2 (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴})
9 posn 5757 . 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 1084  wcel 2099  wral 3057  {csn 4624   class class class wbr 5142   Po wpo 5582   Or wor 5583  Rel wrel 5677
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1790  ax-4 1804  ax-5 1906  ax-6 1964  ax-7 2004  ax-8 2101  ax-9 2109  ax-ext 2699  ax-sep 5293  ax-nul 5300  ax-pr 5423
This theorem depends on definitions:  df-bi 206  df-an 396  df-or 847  df-3or 1086  df-3an 1087  df-tru 1537  df-fal 1547  df-ex 1775  df-sb 2061  df-clab 2706  df-cleq 2720  df-clel 2806  df-ral 3058  df-rex 3067  df-rab 3429  df-v 3472  df-dif 3948  df-un 3950  df-in 3952  df-ss 3962  df-nul 4319  df-if 4525  df-sn 4625  df-pr 4627  df-op 4631  df-br 5143  df-opab 5205  df-po 5584  df-so 5585  df-xp 5678  df-rel 5679
This theorem is referenced by:  wesn  5760
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