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Theorem poirr2 4938
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 4854 . . . 4 Rel ( I ↾ 𝐴)
2 relin2 4665 . . . 4 (Rel ( I ↾ 𝐴) → Rel (𝑅 ∩ ( I ↾ 𝐴)))
31, 2mp1i 10 . . 3 (𝑅 Po 𝐴 → Rel (𝑅 ∩ ( I ↾ 𝐴)))
4 df-br 3937 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)))
5 brin 3987 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
64, 5bitr3i 185 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
7 vex 2692 . . . . . . . . 9 𝑦 ∈ V
87brres 4832 . . . . . . . 8 (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥 I 𝑦𝑥𝐴))
9 poirr 4236 . . . . . . . . . . 11 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
107ideq 4698 . . . . . . . . . . . . 13 (𝑥 I 𝑦𝑥 = 𝑦)
11 breq2 3940 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1210, 11sylbi 120 . . . . . . . . . . . 12 (𝑥 I 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1312notbid 657 . . . . . . . . . . 11 (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
149, 13syl5ibcom 154 . . . . . . . . . 10 ((𝑅 Po 𝐴𝑥𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
1514expimpd 361 . . . . . . . . 9 (𝑅 Po 𝐴 → ((𝑥𝐴𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
1615ancomsd 267 . . . . . . . 8 (𝑅 Po 𝐴 → ((𝑥 I 𝑦𝑥𝐴) → ¬ 𝑥𝑅𝑦))
178, 16syl5bi 151 . . . . . . 7 (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
1817con2d 614 . . . . . 6 (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
19 imnan 680 . . . . . 6 ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2018, 19sylib 121 . . . . 5 (𝑅 Po 𝐴 → ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2120pm2.21d 609 . . . 4 (𝑅 Po 𝐴 → ((𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦) → ⟨𝑥, 𝑦⟩ ∈ ∅))
226, 21syl5bi 151 . . 3 (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
233, 22relssdv 4638 . 2 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅)
24 ss0 3407 . 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 103  wb 104   = wceq 1332  wcel 1481  cin 3074  wss 3075  c0 3367  cop 3534   class class class wbr 3936   I cid 4217   Po wpo 4223  cres 4548  Rel wrel 4551
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4053  ax-pow 4105  ax-pr 4138
This theorem depends on definitions:  df-bi 116  df-3an 965  df-tru 1335  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ral 2422  df-rex 2423  df-v 2691  df-dif 3077  df-un 3079  df-in 3081  df-ss 3088  df-nul 3368  df-pw 3516  df-sn 3537  df-pr 3538  df-op 3540  df-br 3937  df-opab 3997  df-id 4222  df-po 4225  df-xp 4552  df-rel 4553  df-res 4558
This theorem is referenced by: (None)
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