Metamath Proof Explorer < Previous   Next > Nearby theorems Mirrors  >  Home  >  MPE Home  >  Th. List  >  poirr2 Structured version   Visualization version   GIF version

Theorem poirr2 5960
 Description: A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.) (Proof shortened by Peter Mazsa, 2-Oct-2022.)
Assertion
Ref Expression
poirr2 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)

Proof of Theorem poirr2
Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 relres 5856 . . . 4 Rel ( I ↾ 𝐴)
2 relin2 5659 . . . 4 (Rel ( I ↾ 𝐴) → Rel (𝑅 ∩ ( I ↾ 𝐴)))
31, 2mp1i 13 . . 3 (𝑅 Po 𝐴 → Rel (𝑅 ∩ ( I ↾ 𝐴)))
4 df-br 5036 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)))
5 brin 5087 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
64, 5bitr3i 280 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
7 vex 3413 . . . . . . . . 9 𝑦 ∈ V
87brresi 5836 . . . . . . . 8 (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥𝐴𝑥 I 𝑦))
9 poirr 5457 . . . . . . . . . 10 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
107ideq 5697 . . . . . . . . . . . 12 (𝑥 I 𝑦𝑥 = 𝑦)
11 breq2 5039 . . . . . . . . . . . 12 (𝑥 = 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1210, 11sylbi 220 . . . . . . . . . . 11 (𝑥 I 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1312notbid 321 . . . . . . . . . 10 (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
149, 13syl5ibcom 248 . . . . . . . . 9 ((𝑅 Po 𝐴𝑥𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
1514expimpd 457 . . . . . . . 8 (𝑅 Po 𝐴 → ((𝑥𝐴𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
168, 15syl5bi 245 . . . . . . 7 (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
1716con2d 136 . . . . . 6 (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
18 imnan 403 . . . . . 6 ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
1917, 18sylib 221 . . . . 5 (𝑅 Po 𝐴 → ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2019pm2.21d 121 . . . 4 (𝑅 Po 𝐴 → ((𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦) → ⟨𝑥, 𝑦⟩ ∈ ∅))
216, 20syl5bi 245 . . 3 (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
223, 21relssdv 5634 . 2 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅)
23 ss0 4297 . 2 ((𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅ → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)
2422, 23syl 17 1 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)
 Colors of variables: wff setvar class Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   = wceq 1538   ∈ wcel 2111   ∩ cin 3859   ⊆ wss 3860  ∅c0 4227  ⟨cop 4531   class class class wbr 5035   I cid 5432   Po wpo 5444   ↾ cres 5529  Rel wrel 5532 This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5172  ax-nul 5179  ax-pr 5301 This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-dif 3863  df-un 3865  df-in 3867  df-ss 3877  df-nul 4228  df-if 4424  df-sn 4526  df-pr 4528  df-op 4532  df-br 5036  df-opab 5098  df-id 5433  df-po 5446  df-xp 5533  df-rel 5534  df-res 5539 This theorem is referenced by: (None)
 Copyright terms: Public domain W3C validator