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Theorem posng 4439
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 4060 . 2 (𝑅 Po {𝐴} ↔ ∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)))
2 breq2 3795 . . . . . . . . . . 11 (𝑥 = 𝐴 → (𝑦𝑅𝑥𝑦𝑅𝐴))
32anbi2d 445 . . . . . . . . . 10 (𝑥 = 𝐴 → ((𝑧𝑅𝑦𝑦𝑅𝑥) ↔ (𝑧𝑅𝑦𝑦𝑅𝐴)))
4 breq2 3795 . . . . . . . . . 10 (𝑥 = 𝐴 → (𝑧𝑅𝑥𝑧𝑅𝐴))
53, 4imbi12d 227 . . . . . . . . 9 (𝑥 = 𝐴 → (((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥) ↔ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)))
65anbi2d 445 . . . . . . . 8 (𝑥 = 𝐴 → ((¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
76ralsng 3438 . . . . . . 7 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
87ralbidv 2343 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
9 simpl 106 . . . . . . . . . 10 ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝑦)
10 breq2 3795 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑧𝑅𝑦𝑧𝑅𝐴))
119, 10syl5ib 147 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))
1211biantrud 292 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑧𝑅𝑧 ↔ (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴))))
1312bicomd 133 . . . . . . 7 (𝑦 = 𝐴 → ((¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)) ↔ ¬ 𝑧𝑅𝑧))
1413ralsng 3438 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝐴) → 𝑧𝑅𝐴)) ↔ ¬ 𝑧𝑅𝑧))
158, 14bitrd 181 . . . . 5 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝑧𝑅𝑧))
1615ralbidv 2343 . . . 4 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑧))
17 breq12 3796 . . . . . . 7 ((𝑧 = 𝐴𝑧 = 𝐴) → (𝑧𝑅𝑧𝐴𝑅𝐴))
1817anidms 383 . . . . . 6 (𝑧 = 𝐴 → (𝑧𝑅𝑧𝐴𝑅𝐴))
1918notbid 602 . . . . 5 (𝑧 = 𝐴 → (¬ 𝑧𝑅𝑧 ↔ ¬ 𝐴𝑅𝐴))
2019ralsng 3438 . . . 4 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} ¬ 𝑧𝑅𝑧 ↔ ¬ 𝐴𝑅𝐴))
2116, 20bitrd 181 . . 3 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝐴𝑅𝐴))
2221adantl 266 . 2 ((Rel 𝑅𝐴 ∈ V) → (∀𝑧 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑥 ∈ {𝐴} (¬ 𝑧𝑅𝑧 ∧ ((𝑧𝑅𝑦𝑦𝑅𝑥) → 𝑧𝑅𝑥)) ↔ ¬ 𝐴𝑅𝐴))
231, 22syl5bb 185 1 ((Rel 𝑅𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 101  wb 102   = wceq 1259  wcel 1409  wral 2323  Vcvv 2574  {csn 3402   class class class wbr 3791   Po wpo 4058  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-v 2576  df-sbc 2787  df-un 2949  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-po 4060
This theorem is referenced by:  sosng  4440
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