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Theorem poirr2 4811
Description: A partial order relation is irreflexive. (Contributed by Mario Carneiro, 2-Nov-2015.)
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 4728 . . . 4 Rel ( I ↾ 𝐴)
2 relin2 4544 . . . 4 (Rel ( I ↾ 𝐴) → Rel (𝑅 ∩ ( I ↾ 𝐴)))
31, 2mp1i 10 . . 3 (𝑅 Po 𝐴 → Rel (𝑅 ∩ ( I ↾ 𝐴)))
4 df-br 3838 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)))
5 brin 3884 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
64, 5bitr3i 184 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
7 vex 2622 . . . . . . . . 9 𝑦 ∈ V
87brres 4707 . . . . . . . 8 (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥 I 𝑦𝑥𝐴))
9 poirr 4125 . . . . . . . . . . 11 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
107ideq 4576 . . . . . . . . . . . . 13 (𝑥 I 𝑦𝑥 = 𝑦)
11 breq2 3841 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1210, 11sylbi 119 . . . . . . . . . . . 12 (𝑥 I 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1312notbid 627 . . . . . . . . . . 11 (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
149, 13syl5ibcom 153 . . . . . . . . . 10 ((𝑅 Po 𝐴𝑥𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
1514expimpd 355 . . . . . . . . 9 (𝑅 Po 𝐴 → ((𝑥𝐴𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
1615ancomsd 265 . . . . . . . 8 (𝑅 Po 𝐴 → ((𝑥 I 𝑦𝑥𝐴) → ¬ 𝑥𝑅𝑦))
178, 16syl5bi 150 . . . . . . 7 (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
1817con2d 589 . . . . . 6 (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
19 imnan 659 . . . . . 6 ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2018, 19sylib 120 . . . . 5 (𝑅 Po 𝐴 → ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2120pm2.21d 584 . . . 4 (𝑅 Po 𝐴 → ((𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦) → ⟨𝑥, 𝑦⟩ ∈ ∅))
226, 21syl5bi 150 . . 3 (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
233, 22relssdv 4518 . 2 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅)
24 ss0 3320 . 2 ((𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅ → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)
2523, 24syl 14 1 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)
Colors of variables: wff set class
Syntax hints:  ¬ wn 3  wi 4  wa 102  wb 103   = wceq 1289  wcel 1438  cin 2996  wss 2997  c0 3284  cop 3444   class class class wbr 3837   I cid 4106   Po wpo 4112  cres 4430  Rel wrel 4433
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 579  ax-in2 580  ax-io 665  ax-5 1381  ax-7 1382  ax-gen 1383  ax-ie1 1427  ax-ie2 1428  ax-8 1440  ax-10 1441  ax-11 1442  ax-i12 1443  ax-bndl 1444  ax-4 1445  ax-14 1450  ax-17 1464  ax-i9 1468  ax-ial 1472  ax-i5r 1473  ax-ext 2070  ax-sep 3949  ax-pow 4001  ax-pr 4027
This theorem depends on definitions:  df-bi 115  df-3an 926  df-tru 1292  df-nf 1395  df-sb 1693  df-eu 1951  df-mo 1952  df-clab 2075  df-cleq 2081  df-clel 2084  df-nfc 2217  df-ral 2364  df-rex 2365  df-v 2621  df-dif 2999  df-un 3001  df-in 3003  df-ss 3010  df-nul 3285  df-pw 3427  df-sn 3447  df-pr 3448  df-op 3450  df-br 3838  df-opab 3892  df-id 4111  df-po 4114  df-xp 4434  df-rel 4435  df-res 4440
This theorem is referenced by: (None)
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