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Theorem poirr2 4744
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 4666 . . . 4 Rel ( I ↾ 𝐴)
2 relin2 4483 . . . 4 (Rel ( I ↾ 𝐴) → Rel (𝑅 ∩ ( I ↾ 𝐴)))
31, 2mp1i 10 . . 3 (𝑅 Po 𝐴 → Rel (𝑅 ∩ ( I ↾ 𝐴)))
4 df-br 3792 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)))
5 brin 3838 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
64, 5bitr3i 179 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
7 vex 2577 . . . . . . . . 9 𝑦 ∈ V
87brres 4645 . . . . . . . 8 (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥 I 𝑦𝑥𝐴))
9 poirr 4071 . . . . . . . . . . 11 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
107ideq 4515 . . . . . . . . . . . . 13 (𝑥 I 𝑦𝑥 = 𝑦)
11 breq2 3795 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1210, 11sylbi 118 . . . . . . . . . . . 12 (𝑥 I 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1312notbid 602 . . . . . . . . . . 11 (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
149, 13syl5ibcom 148 . . . . . . . . . 10 ((𝑅 Po 𝐴𝑥𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
1514expimpd 349 . . . . . . . . 9 (𝑅 Po 𝐴 → ((𝑥𝐴𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
1615ancomsd 260 . . . . . . . 8 (𝑅 Po 𝐴 → ((𝑥 I 𝑦𝑥𝐴) → ¬ 𝑥𝑅𝑦))
178, 16syl5bi 145 . . . . . . 7 (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
1817con2d 564 . . . . . 6 (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
19 imnan 634 . . . . . 6 ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2018, 19sylib 131 . . . . 5 (𝑅 Po 𝐴 → ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2120pm2.21d 559 . . . 4 (𝑅 Po 𝐴 → ((𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦) → ⟨𝑥, 𝑦⟩ ∈ ∅))
226, 21syl5bi 145 . . 3 (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
233, 22relssdv 4459 . 2 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅)
24 ss0 3284 . 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 101  wb 102   = wceq 1259  wcel 1409  cin 2943  wss 2944  c0 3251  cop 3405   class class class wbr 3791   I cid 4052   Po wpo 4058  cres 4374  Rel wrel 4377
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 103  ax-ia2 104  ax-ia3 105  ax-in1 554  ax-in2 555  ax-io 640  ax-5 1352  ax-7 1353  ax-gen 1354  ax-ie1 1398  ax-ie2 1399  ax-8 1411  ax-10 1412  ax-11 1413  ax-i12 1414  ax-bndl 1415  ax-4 1416  ax-14 1421  ax-17 1435  ax-i9 1439  ax-ial 1443  ax-i5r 1444  ax-ext 2038  ax-sep 3902  ax-pow 3954  ax-pr 3971
This theorem depends on definitions:  df-bi 114  df-3an 898  df-tru 1262  df-nf 1366  df-sb 1662  df-eu 1919  df-mo 1920  df-clab 2043  df-cleq 2049  df-clel 2052  df-nfc 2183  df-ral 2328  df-rex 2329  df-v 2576  df-dif 2947  df-un 2949  df-in 2951  df-ss 2958  df-nul 3252  df-pw 3388  df-sn 3408  df-pr 3409  df-op 3411  df-br 3792  df-opab 3846  df-id 4057  df-po 4060  df-xp 4378  df-rel 4379  df-res 4384
This theorem is referenced by: (None)
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