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Theorem sosn 5100
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 4141 . . . . . 6 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
2 elsni 4141 . . . . . . 7 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
32eqcomd 2615 . . . . . 6 (𝑦 ∈ {𝐴} → 𝐴 = 𝑦)
41, 3sylan9eq 2663 . . . . 5 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → 𝑥 = 𝑦)
543mix2d 1229 . . . 4 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥))
65rgen2a 2959 . . 3 𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)
7 df-so 4949 . . 3 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴} (𝑥𝑅𝑦𝑥 = 𝑦𝑦𝑅𝑥)))
86, 7mpbiran2 955 . 2 (𝑅 Or {𝐴} ↔ 𝑅 Po {𝐴})
9 posn 5099 . 2 (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
108, 9syl5bb 270 1 (Rel 𝑅 → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 194  wa 382  w3o 1029  wcel 1976  wral 2895  {csn 4124   class class class wbr 4577   Po wpo 4946   Or wor 4947  Rel wrel 5032
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1712  ax-4 1727  ax-5 1826  ax-6 1874  ax-7 1921  ax-9 1985  ax-10 2005  ax-11 2020  ax-12 2033  ax-13 2233  ax-ext 2589  ax-sep 4703  ax-nul 4711  ax-pr 4827
This theorem depends on definitions:  df-bi 195  df-or 383  df-an 384  df-3or 1031  df-3an 1032  df-tru 1477  df-ex 1695  df-nf 1700  df-sb 1867  df-clab 2596  df-cleq 2602  df-clel 2605  df-nfc 2739  df-ral 2900  df-rex 2901  df-rab 2904  df-v 3174  df-sbc 3402  df-dif 3542  df-un 3544  df-in 3546  df-ss 3553  df-nul 3874  df-if 4036  df-sn 4125  df-pr 4127  df-op 4131  df-br 4578  df-opab 4638  df-po 4948  df-so 4949  df-xp 5033  df-rel 5034
This theorem is referenced by:  wesn  5102
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