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Theorem posng 4510
Description: Partial ordering of a singleton. (Contributed by Jim Kingdon, 5-Dec-2018.)
Assertion
Ref Expression
posng ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem posng
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-po 4123 . 2 (𝑅 Po {𝐴} ↔ ∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)))
2 breq2 3849 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
32anbi2d 452 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑧𝑅𝑦𝑦𝑅𝑥) ↔ (𝑧𝑅𝑦𝑦𝑅𝐴)))
4 breq2 3849 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑧𝑅𝑥𝑧𝑅𝐴))
53, 4imbi12d 232 . . . . . . . . 9 (𝑥 = 𝐴 → (((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥) ↔ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)))
65anbi2d 452 . . . . . . . 8 (𝑥 = 𝐴 → ((¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
76ralsng 3483 . . . . . . 7 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
87ralbidv 2380 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
9 simpl 107 . . . . . . . . . 10 ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝑦)
10 breq2 3849 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑅𝑦𝑧𝑅𝐴))
119, 10syl5ib 152 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))
1211biantrud 298 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑧 ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
1312bicomd 139 . . . . . . 7 (𝑦 = 𝐴 → ((¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)) ↔ ¬ 𝑧𝑅𝑧))
1413ralsng 3483 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)) ↔ ¬ 𝑧𝑅𝑧))
158, 14bitrd 186 . . . . 5 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝑧𝑅𝑧))
1615ralbidv 2380 . . . 4 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑧))
17 breq12 3850 . . . . . . 7 ((𝑧 = 𝐴𝑧 = 𝐴) → (𝑧𝑅𝑧𝐴𝑅𝐴))
1817anidms 389 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑅𝑧𝐴𝑅𝐴))
1918notbid 627 . . . . 5 (𝑧 = 𝐴 → (¬ 𝑧𝑅𝑧 ↔ ¬ 𝐴𝑅𝐴))
2019ralsng 3483 . . . 4 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑧 ↔ ¬ 𝐴𝑅𝐴))
2116, 20bitrd 186 . . 3 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝐴𝑅𝐴))
2221adantl 271 . 2 ((Rel 𝑅𝐴 ∈ V) → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝐴𝑅𝐴))
231, 22syl5bb 190 1 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  wral 2359  Vcvv 2619  {csn 3446   class class class wbr 3845   Po wpo 4121  Rel wrel 4443
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 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-v 2621  df-sbc 2841  df-un 3003  df-sn 3452  df-pr 3453  df-op 3455  df-br 3846  df-po 4123
This theorem is referenced by:  sosng  4511
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