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Theorem posng 4606
 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 4213 . 2 (𝑅 Po {𝐴} ↔ ∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)))
2 breq2 3928 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
32anbi2d 459 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑧𝑅𝑦𝑦𝑅𝑥) ↔ (𝑧𝑅𝑦𝑦𝑅𝐴)))
4 breq2 3928 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑧𝑅𝑥𝑧𝑅𝐴))
53, 4imbi12d 233 . . . . . . . . 9 (𝑥 = 𝐴 → (((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥) ↔ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)))
65anbi2d 459 . . . . . . . 8 (𝑥 = 𝐴 → ((¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
76ralsng 3559 . . . . . . 7 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
87ralbidv 2435 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
9 simpl 108 . . . . . . . . . 10 ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝑦)
10 breq2 3928 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑅𝑦𝑧𝑅𝐴))
119, 10syl5ib 153 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))
1211biantrud 302 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑧 ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
1312bicomd 140 . . . . . . 7 (𝑦 = 𝐴 → ((¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)) ↔ ¬ 𝑧𝑅𝑧))
1413ralsng 3559 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)) ↔ ¬ 𝑧𝑅𝑧))
158, 14bitrd 187 . . . . 5 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝑧𝑅𝑧))
1615ralbidv 2435 . . . 4 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑧))
17 breq12 3929 . . . . . . 7 ((𝑧 = 𝐴𝑧 = 𝐴) → (𝑧𝑅𝑧𝐴𝑅𝐴))
1817anidms 394 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑅𝑧𝐴𝑅𝐴))
1918notbid 656 . . . . 5 (𝑧 = 𝐴 → (¬ 𝑧𝑅𝑧 ↔ ¬ 𝐴𝑅𝐴))
2019ralsng 3559 . . . 4 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑧 ↔ ¬ 𝐴𝑅𝐴))
2116, 20bitrd 187 . . 3 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝐴𝑅𝐴))
2221adantl 275 . 2 ((Rel 𝑅𝐴 ∈ V) → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝐴𝑅𝐴))
231, 22syl5bb 191 1 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
 Colors of variables: wff set class Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   = wceq 1331   ∈ wcel 1480  ∀wral 2414  Vcvv 2681  {csn 3522   class class class wbr 3924   Po wpo 4211  Rel wrel 4539 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 2119 This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-v 2683  df-sbc 2905  df-un 3070  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-po 4213 This theorem is referenced by:  sosng  4607
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