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Theorem posn 5344
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 5202 . . . . . 6 𝑅 Po ∅
2 snprc 4397 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
3 poeq2 5191 . . . . . . 7 ({𝐴} = ∅ → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
42, 3sylbi 207 . . . . . 6 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ 𝑅 Po ∅))
51, 4mpbiri 248 . . . . 5 𝐴 ∈ V → 𝑅 Po {𝐴})
65adantl 473 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Po {𝐴})
7 brrelex 5313 . . . . 5 ((Rel 𝑅𝐴𝑅𝐴) → 𝐴 ∈ V)
87stoic1a 1846 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → ¬ 𝐴𝑅𝐴)
96, 82thd 255 . . 3 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
109ex 449 . 2 (Rel 𝑅 → (¬ 𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴)))
11 df-po 5187 . . 3 (𝑅 Po {𝐴} ↔ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12 breq2 4808 . . . . . . . . . . 11 (𝑧 = 𝐴 → (𝑦𝑅𝑧𝑦𝑅𝐴))
1312anbi2d 742 . . . . . . . . . 10 (𝑧 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐴)))
14 breq2 4808 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝑥𝑅𝑧𝑥𝑅𝐴))
1513, 14imbi12d 333 . . . . . . . . 9 (𝑧 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)))
1615anbi2d 742 . . . . . . . 8 (𝑧 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
1716ralsng 4362 . . . . . . 7 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
1817ralbidv 3124 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
19 simpl 474 . . . . . . . . . 10 ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝑦)
20 breq2 4808 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥𝑅𝑦𝑥𝑅𝐴))
2119, 20syl5ib 234 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))
2221biantrud 529 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
2322bicomd 213 . . . . . . 7 (𝑦 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
2423ralsng 4362 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
2518, 24bitrd 268 . . . . 5 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝑥𝑅𝑥))
2625ralbidv 3124 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥))
27 breq12 4809 . . . . . . 7 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
2827anidms 680 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
2928notbid 307 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
3029ralsng 4362 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
3126, 30bitrd 268 . . 3 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝐴𝑅𝐴))
3211, 31syl5bb 272 . 2 (𝐴 ∈ V → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
3310, 32pm2.61d2 172 1 (Rel 𝑅 → (𝑅 Po {𝐴} ↔ ¬ 𝐴𝑅𝐴))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wa 383   = wceq 1632  wcel 2139  wral 3050  Vcvv 3340  c0 4058  {csn 4321   class class class wbr 4804   Po wpo 5185  Rel wrel 5271
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1871  ax-4 1886  ax-5 1988  ax-6 2054  ax-7 2090  ax-9 2148  ax-10 2168  ax-11 2183  ax-12 2196  ax-13 2391  ax-ext 2740  ax-sep 4933  ax-nul 4941  ax-pr 5055
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1074  df-tru 1635  df-ex 1854  df-nf 1859  df-sb 2047  df-clab 2747  df-cleq 2753  df-clel 2756  df-nfc 2891  df-ral 3055  df-rex 3056  df-rab 3059  df-v 3342  df-sbc 3577  df-dif 3718  df-un 3720  df-in 3722  df-ss 3729  df-nul 4059  df-if 4231  df-sn 4322  df-pr 4324  df-op 4328  df-br 4805  df-opab 4865  df-po 5187  df-xp 5272  df-rel 5273
This theorem is referenced by:  sosn  5345
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