Theorem poirr2 6029

Description: A partial order 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.

Step	Hyp	Ref	Expression
1		relres 5920	. . . 4 ⊢ Rel ( I ↾ 𝐴)
2		relin2 5723	. . . 4 ⊢ (Rel ( I ↾ 𝐴) → Rel (𝑅 ∩ ( I ↾ 𝐴)))
3	1, 2	mp1i 13	. . 3 ⊢ (𝑅 Po 𝐴 → Rel (𝑅 ∩ ( I ↾ 𝐴)))
4		df-br 5075	. . . . 5 ⊢ (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)))
5		brin 5126	. . . . 5 ⊢ (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
6	4, 5	bitr3i 276	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
7		vex 3436	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	7	brresi 5900	. . . . . . . 8 ⊢ (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 I 𝑦))
9		poirr 5515	. . . . . . . . . 10 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
10	7	ideq 5761	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
11		breq2 5078	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑦))
12	10, 11	sylbi 216	. . . . . . . . . . 11 ⊢ (𝑥 I 𝑦 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑦))
13	12	notbid 318	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
14	9, 13	syl5ibcom 244	. . . . . . . . 9 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
15	14	expimpd 454	. . . . . . . 8 ⊢ (𝑅 Po 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
16	8, 15	syl5bi 241	. . . . . . 7 ⊢ (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
17	16	con2d 134	. . . . . 6 ⊢ (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
18		imnan 400	. . . . . 6 ⊢ ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
19	17, 18	sylib 217	. . . . 5 ⊢ (𝑅 Po 𝐴 → ¬ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
20	19	pm2.21d 121	. . . 4 ⊢ (𝑅 Po 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦) → ⟨𝑥, 𝑦⟩ ∈ ∅))
21	6, 20	syl5bi 241	. . 3 ⊢ (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
22	3, 21	relssdv 5698	. 2 ⊢ (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅)
23		ss0 4332	. 2 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅ → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)
24	22, 23	syl 17	1 ⊢ (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 205   ∧ wa 396   = wceq 1539   ∈ wcel 2106   ∩ cin 3886   ⊆ wss 3887  ∅c0 4256  ⟨cop 4567   class class class wbr 5074   I cid 5488   Po wpo 5501   ↾ cres 5591  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709  ax-sep 5223  ax-nul 5230  ax-pr 5352

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 845  df-3an 1088  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-ral 3069  df-rex 3070  df-rab 3073  df-v 3434  df-dif 3890  df-un 3892  df-in 3894  df-ss 3904  df-nul 4257  df-if 4460  df-sn 4562  df-pr 4564  df-op 4568  df-br 5075  df-opab 5137  df-id 5489  df-po 5503  df-xp 5595  df-rel 5596  df-res 5601

This theorem is referenced by: (None)