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Theorem sosng 4696
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 4310 . . 3 (𝑅 Or {𝐴} → 𝑅 Po {𝐴})
2 posng 4695 . . 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 615 . . . . . . . . 9 𝐴𝑅𝐴 → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
65adantr 276 . . . . . . . 8 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → (𝐴𝑅𝐴 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
7 elsni 3609 . . . . . . . . . . 11 (𝑥 ∈ {𝐴} → 𝑥 = 𝐴)
8 elsni 3609 . . . . . . . . . . 11 (𝑦 ∈ {𝐴} → 𝑦 = 𝐴)
97, 8breqan12d 4016 . . . . . . . . . 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 2551 . . . . . 6 ((¬ 𝐴𝑅𝐴 ∧ (𝑥 ∈ {𝐴} ∧ 𝑦 ∈ {𝐴})) → ∀𝑧 ∈ {𝐴} (𝑥𝑅𝑦 → (𝑥𝑅𝑧𝑧𝑅𝑦)))
1413ralrimivva 2559 . . . . 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 4294 . . . 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 708  wcel 2148  wral 2455  Vcvv 2737  {csn 3591   class class class wbr 4000   Po wpo 4291   Or wor 4292  Rel wrel 4628
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ral 2460  df-v 2739  df-sbc 2963  df-un 3133  df-sn 3597  df-pr 3598  df-op 3600  df-br 4001  df-po 4293  df-iso 4294
This theorem is referenced by: (None)
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