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Theorem sosng 4540
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 4164 . . 3 (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2 posng 4539 . . 3 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
31, 2syl5ib 153 . 2 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Or {𝐴} → ¬ 𝐴𝑅𝐴))
42biimpar 292 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → 𝑅 Po {𝐴})
5 ax-in2 583 . . . . . . . . 9 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
65adantr 271 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
7 elsni 3484 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8 elsni 3484 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
97, 8breqan12d 3882 . . . . . . . . . 10 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → (𝑥𝑅𝑦𝐴𝑅𝐴))
109imbi1d 230 . . . . . . . . 9 ((𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴}) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
1110adantl 272 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ((𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)) ↔ (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
126, 11mpbird 166 . . . . . . 7 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1312ralrimivw 2459 . . . . . 6 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1413ralrimivva 2467 . . . . 5 𝐴𝑅𝐴 → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1514adantl 272 . . . 4 (((Rel 𝑅𝐴 ∈ V) ∧ ¬ 𝐴𝑅𝐴) → ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
16 df-iso 4148 . . . 4 (𝑅 Or {𝐴} ↔ (𝑅 Po {𝐴} ∧ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦))))
174, 15, 16sylanbrc 409 . . 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 667  wcel 1445  wral 2370  Vcvv 2633  {csn 3466   class class class wbr 3867   Po wpo 4145   Or wor 4146  Rel wrel 4472
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 582  ax-in2 583  ax-io 668  ax-5 1388  ax-7 1389  ax-gen 1390  ax-ie1 1434  ax-ie2 1435  ax-8 1447  ax-10 1448  ax-11 1449  ax-i12 1450  ax-bndl 1451  ax-4 1452  ax-17 1471  ax-i9 1475  ax-ial 1479  ax-i5r 1480  ax-ext 2077
This theorem depends on definitions:  df-bi 116  df-3an 929  df-tru 1299  df-nf 1402  df-sb 1700  df-clab 2082  df-cleq 2088  df-clel 2091  df-nfc 2224  df-ral 2375  df-v 2635  df-sbc 2855  df-un 3017  df-sn 3472  df-pr 3473  df-op 3475  df-br 3868  df-po 4147  df-iso 4148
This theorem is referenced by: (None)
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