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Theorem posn 5672
Description: Partial ordering of a singleton. (Contributed by NM, 27-Apr-2009.) (Revised by Mario Carneiro, 23-Apr-2015.)
Assertion
Ref Expression
posn (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))

Proof of Theorem posn
Dummy variables 𝑥 𝑦 𝑧 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 po0 5520 . . . . . 6 𝑅 Po ∅
2 snprc 4653 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
3 poeq2 5507 . . . . . . 7 ({𝐴} = ∅ → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
42, 3sylbi 216 . . . . . 6 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
51, 4mpbiri 257 . . . . 5 𝐴 ∈ V → 𝑅 Po {𝐴})
65adantl 482 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Po {𝐴})
7 brrelex1 5640 . . . . 5 ((Rel 𝑅𝐴𝑅𝐴) → 𝐴 ∈ V)
87stoic1a 1775 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
96, 82thd 264 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
109ex 413 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11 df-po 5503 . . 3 (𝑅 Po {𝐴} ↔ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12 breq2 5078 . . . . . . . . . . 11 (𝑧 = 𝐴 → (𝑦𝑅𝑧𝑦𝑅𝐴))
1312anbi2d 629 . . . . . . . . . 10 (𝑧 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐴)))
14 breq2 5078 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝑥𝑅𝑧𝑥𝑅𝐴))
1513, 14imbi12d 345 . . . . . . . . 9 (𝑧 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)))
1615anbi2d 629 . . . . . . . 8 (𝑧 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
1716ralsng 4609 . . . . . . 7 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
1817ralbidv 3112 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
19 simpl 483 . . . . . . . . . 10 ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝑦)
20 breq2 5078 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥𝑅𝑦𝑥𝑅𝐴))
2119, 20syl5ib 243 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))
2221biantrud 532 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
2322bicomd 222 . . . . . . 7 (𝑦 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
2423ralsng 4609 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
2518, 24bitrd 278 . . . . 5 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝑥𝑅𝑥))
2625ralbidv 3112 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥))
27 breq12 5079 . . . . . . 7 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
2827anidms 567 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
2928notbid 318 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
3029ralsng 4609 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
3126, 30bitrd 278 . . 3 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝐴𝑅𝐴))
3211, 31bitrid 282 . 2 (𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
3310, 32pm2.61d2 181 1 (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wa 396   = wceq 1539  wcel 2106  wral 3064  Vcvv 3432  c0 4256  {csn 4561   class class class wbr 5074   Po wpo 5501  Rel wrel 5594
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352
This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-po 5503  df-xp 5595  df-rel 5596
This theorem is referenced by:  sosn  5673
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