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Theorem sosng 4612
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 4235 . . 3 (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2 posng 4611 . . 3 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
31, 2syl5ib 153 . 2 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} → ¬ 𝐴𝑅𝐴))
42biimpar 295 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Po {𝐴})
5 ax-in2 604 . . . . . . . . 9 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
65adantr 274 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
7 elsni 3545 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8 elsni 3545 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
97, 8breqan12d 3945 . . . . . . . . . 10 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝐴𝑅𝐴))
109imbi1d 230 . . . . . . . . 9 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1110adantl 275 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
126, 11mpbird 166 . . . . . . 7 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1312ralrimivw 2506 . . . . . 6 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1413ralrimivva 2514 . . . . 5 𝐴𝑅𝐴 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1514adantl 275 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
16 df-iso 4219 . . . 4 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
174, 15, 16sylanbrc 413 . . 3 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Or {𝐴})
1817ex 114 . 2 ((Rel 𝑅𝐴 ∈ V) → (¬ 𝐴𝑅𝐴𝑅 Or {𝐴}))
193, 18impbid 128 1 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 103  wb 104  wo 697  wcel 1480  wral 2416  Vcvv 2686  {csn 3527   class class class wbr 3929   Po wpo 4216   Or wor 4217  Rel wrel 4544
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-v 2688  df-sbc 2910  df-un 3075  df-sn 3533  df-pr 3534  df-op 3536  df-br 3930  df-po 4218  df-iso 4219
This theorem is referenced by: (None)
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