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Theorem posn 5221
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 5079 . . . . . 6 𝑅 Po ∅
2 snprc 4285 . . . . . . 7 𝐴 ∈ V ↔ {𝐴} = ∅)
3 poeq2 5068 . . . . . . 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 481 . . . 4 ((Rel 𝑅 ∧ ¬ 𝐴 ∈ V) → 𝑅 Po {𝐴})
7 brrelex 5190 . . . . 5 ((Rel 𝑅𝐴𝑅𝐴) → 𝐴 ∈ V)
87stoic1a 1737 . . . 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 5064 . . 3 (𝑅 Po {𝐴} ↔ ∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)))
12 breq2 4689 . . . . . . . . . . 11 (𝑧 = 𝐴 → (𝑦𝑅𝑧𝑦𝑅𝐴))
1312anbi2d 740 . . . . . . . . . 10 (𝑧 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝑧) ↔ (𝑥𝑅𝑦𝑦𝑅𝐴)))
14 breq2 4689 . . . . . . . . . 10 (𝑧 = 𝐴 → (𝑥𝑅𝑧𝑥𝑅𝐴))
1513, 14imbi12d 333 . . . . . . . . 9 (𝑧 = 𝐴 → (((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧) ↔ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)))
1615anbi2d 740 . . . . . . . 8 (𝑧 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
1716ralsng 4250 . . . . . . 7 (𝐴 ∈ V → (∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
1817ralbidv 3015 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
19 simpl 472 . . . . . . . . . 10 ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝑦)
20 breq2 4689 . . . . . . . . . 10 (𝑦 = 𝐴 → (𝑥𝑅𝑦𝑥𝑅𝐴))
2119, 20syl5ib 234 . . . . . . . . 9 (𝑦 = 𝐴 → ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))
2221biantrud 527 . . . . . . . 8 (𝑦 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴))))
2322bicomd 213 . . . . . . 7 (𝑦 = 𝐴 → ((¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
2423ralsng 4250 . . . . . 6 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝐴) → 𝑥𝑅𝐴)) ↔ ¬ 𝑥𝑅𝑥))
2518, 24bitrd 268 . . . . 5 (𝐴 ∈ V → (∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ¬ 𝑥𝑅𝑥))
2625ralbidv 3015 . . . 4 (𝐴 ∈ V → (∀𝑥 ∈ {𝐴}∀𝑦 ∈ {𝐴}∀𝑧 ∈ {𝐴} (¬ 𝑥𝑅𝑥 ∧ ((𝑥𝑅𝑦𝑦𝑅𝑧) → 𝑥𝑅𝑧)) ↔ ∀𝑥 ∈ {𝐴} ¬ 𝑥𝑅𝑥))
27 breq12 4690 . . . . . . 7 ((𝑥 = 𝐴𝑥 = 𝐴) → (𝑥𝑅𝑥𝐴𝑅𝐴))
2827anidms 678 . . . . . 6 (𝑥 = 𝐴 → (𝑥𝑅𝑥𝐴𝑅𝐴))
2928notbid 307 . . . . 5 (𝑥 = 𝐴 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝐴𝑅𝐴))
3029ralsng 4250 . . . 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 1523  wcel 2030  wral 2941  Vcvv 3231  c0 3948  {csn 4210   class class class wbr 4685   Po wpo 5062  Rel wrel 5148
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1762  ax-4 1777  ax-5 1879  ax-6 1945  ax-7 1981  ax-9 2039  ax-10 2059  ax-11 2074  ax-12 2087  ax-13 2282  ax-ext 2631  ax-sep 4814  ax-nul 4822  ax-pr 4936
This theorem depends on definitions:  df-bi 197  df-or 384  df-an 385  df-3an 1056  df-tru 1526  df-ex 1745  df-nf 1750  df-sb 1938  df-clab 2638  df-cleq 2644  df-clel 2647  df-nfc 2782  df-ral 2946  df-rex 2947  df-rab 2950  df-v 3233  df-sbc 3469  df-dif 3610  df-un 3612  df-in 3614  df-ss 3621  df-nul 3949  df-if 4120  df-sn 4211  df-pr 4213  df-op 4217  df-br 4686  df-opab 4746  df-po 5064  df-xp 5149  df-rel 5150
This theorem is referenced by:  sosn  5222
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