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Theorem sosng 4717
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 4331 . . 3 (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2 posng 4716 . . 3 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
31, 2imbitrid 154 . 2 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} → ¬ 𝐴𝑅𝐴))
42biimpar 297 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Po {𝐴})
5 ax-in2 616 . . . . . . . . 9 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
65adantr 276 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
7 elsni 3625 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8 elsni 3625 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
97, 8breqan12d 4034 . . . . . . . . . 10 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝐴𝑅𝐴))
109imbi1d 231 . . . . . . . . 9 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1110adantl 277 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
126, 11mpbird 167 . . . . . . 7 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1312ralrimivw 2564 . . . . . 6 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1413ralrimivva 2572 . . . . 5 𝐴𝑅𝐴 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1514adantl 277 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
16 df-iso 4315 . . . 4 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
174, 15, 16sylanbrc 417 . . 3 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Or {𝐴})
1817ex 115 . 2 ((Rel 𝑅𝐴 ∈ V) → (¬ 𝐴𝑅𝐴𝑅 Or {𝐴}))
193, 18impbid 129 1 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 104  wb 105  wo 709  wcel 2160  wral 2468  Vcvv 2752  {csn 3607   class class class wbr 4018   Po wpo 4312   Or wor 4313  Rel wrel 4649
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2171
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2176  df-cleq 2182  df-clel 2185  df-nfc 2321  df-ral 2473  df-v 2754  df-sbc 2978  df-un 3148  df-sn 3613  df-pr 3614  df-op 3616  df-br 4019  df-po 4314  df-iso 4315
This theorem is referenced by: (None)
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