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Theorem sosng 4459
Description: Strict linear ordering on a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
Assertion
Ref Expression
sosng ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem sosng
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 sopo 4096 . . 3 (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2 posng 4458 . . 3 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
31, 2syl5ib 152 . 2 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} → ¬ 𝐴𝑅𝐴))
42biimpar 291 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Po {𝐴})
5 ax-in2 578 . . . . . . . . 9 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
65adantr 270 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
7 elsni 3434 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8 elsni 3434 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
97, 8breqan12d 3820 . . . . . . . . . 10 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝐴𝑅𝐴))
109imbi1d 229 . . . . . . . . 9 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1110adantl 271 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
126, 11mpbird 165 . . . . . . 7 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1312ralrimivw 2440 . . . . . 6 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1413ralrimivva 2448 . . . . 5 𝐴𝑅𝐴 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1514adantl 271 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
16 df-iso 4080 . . . 4 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
174, 15, 16sylanbrc 408 . . 3 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Or {𝐴})
1817ex 113 . 2 ((Rel 𝑅𝐴 ∈ V) → (¬ 𝐴𝑅𝐴𝑅 Or {𝐴}))
193, 18impbid 127 1 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103  wo 662  wcel 1434  wral 2353  Vcvv 2610  {csn 3416   class class class wbr 3805   Po wpo 4077   Or wor 4078  Rel wrel 4396
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2065
This theorem depends on definitions:  df-bi 115  df-3an 922  df-tru 1288  df-nf 1391  df-sb 1688  df-clab 2070  df-cleq 2076  df-clel 2079  df-nfc 2212  df-ral 2358  df-v 2612  df-sbc 2825  df-un 2986  df-sn 3422  df-pr 3423  df-op 3425  df-br 3806  df-po 4079  df-iso 4080
This theorem is referenced by: (None)
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