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Theorem poirr2 4717
 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 4639 . . . 4 Rel ( I ↾ 𝐴)
2 relin2 4456 . . . 4 (Rel ( I ↾ 𝐴) → Rel (𝑅 ∩ ( I ↾ 𝐴)))
31, 2mp1i 10 . . 3 (𝑅 Po 𝐴 → Rel (𝑅 ∩ ( I ↾ 𝐴)))
4 df-br 3765 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)))
5 brin 3811 . . . . 5 (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
64, 5bitr3i 175 . . . 4 (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
7 vex 2560 . . . . . . . . 9 𝑦 ∈ V
87brres 4618 . . . . . . . 8 (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥 I 𝑦𝑥𝐴))
9 poirr 4044 . . . . . . . . . . 11 ((𝑅 Po 𝐴𝑥𝐴) → ¬ 𝑥𝑅𝑥)
107ideq 4488 . . . . . . . . . . . . 13 (𝑥 I 𝑦𝑥 = 𝑦)
11 breq2 3768 . . . . . . . . . . . . 13 (𝑥 = 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1210, 11sylbi 114 . . . . . . . . . . . 12 (𝑥 I 𝑦 → (𝑥𝑅𝑥𝑥𝑅𝑦))
1312notbid 592 . . . . . . . . . . 11 (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
149, 13syl5ibcom 144 . . . . . . . . . 10 ((𝑅 Po 𝐴𝑥𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
1514expimpd 345 . . . . . . . . 9 (𝑅 Po 𝐴 → ((𝑥𝐴𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
1615ancomsd 256 . . . . . . . 8 (𝑅 Po 𝐴 → ((𝑥 I 𝑦𝑥𝐴) → ¬ 𝑥𝑅𝑦))
178, 16syl5bi 141 . . . . . . 7 (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
1817con2d 554 . . . . . 6 (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
19 imnan 624 . . . . . 6 ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2018, 19sylib 127 . . . . 5 (𝑅 Po 𝐴 → ¬ (𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦))
2120pm2.21d 549 . . . 4 (𝑅 Po 𝐴 → ((𝑥𝑅𝑦𝑥( I ↾ 𝐴)𝑦) → ⟨𝑥, 𝑦⟩ ∈ ∅))
226, 21syl5bi 141 . . 3 (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
233, 22relssdv 4432 . 2 (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅)
24 ss0 3257 . 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 97   ↔ wb 98   = wceq 1243   ∈ wcel 1393   ∩ cin 2916   ⊆ wss 2917  ∅c0 3224  ⟨cop 3378   class class class wbr 3764   I cid 4025   Po wpo 4031   ↾ cres 4347  Rel wrel 4350 This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944 This theorem depends on definitions:  df-bi 110  df-3an 887  df-tru 1246  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ral 2311  df-rex 2312  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-nul 3225  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-br 3765  df-opab 3819  df-id 4030  df-po 4033  df-xp 4351  df-rel 4352  df-res 4357 This theorem is referenced by: (None)
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