Theorem poirr2 6080

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 5963	. . . 4 ⊢ Rel ( I ↾ 𝐴)
2		relin2 5758	. . . 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 279	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
7		vex 3437	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	7	brresi 5946	. . . . . . . 8 ⊢ (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥 ∈ 𝐴 ∧ 𝑥 I 𝑦))
9		poirr 5540	. . . . . . . . . 10 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
10	7	ideq 5796	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
11		breq2 5078	. . . . . . . . . . . 12 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑦))
12	10, 11	sylbi 219	. . . . . . . . . . 11 ⊢ (𝑥 I 𝑦 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑦))
13	12	notbid 320	. . . . . . . . . 10 ⊢ (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
14	9, 13	syl5ibcom 247	. . . . . . . . 9 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
15	14	expimpd 455	. . . . . . . 8 ⊢ (𝑅 Po 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
16	8, 15	biimtrid 244	. . . . . . 7 ⊢ (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
17	16	con2d 134	. . . . . 6 ⊢ (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
18		imnan 401	. . . . . 6 ⊢ ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
19	17, 18	sylib 220	. . . . 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	biimtrid 244	. . 3 ⊢ (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
22	3, 21	relssdv 5733	. 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 208   ∧ wa 397   = wceq 1548   ∈ wcel 2121   ∩ cin 3883   ⊆ wss 3884  ∅c0 4263  ⟨cop 4563   class class class wbr 5074   I cid 5514   Po wpo 5526   ↾ cres 5622  Rel wrel 5625

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1975  ax-7 2016  ax-8 2123  ax-9 2131  ax-ext 2713  ax-sep 5220  ax-pr 5364

This theorem depends on definitions:  df-bi 209  df-an 398  df-or 855  df-3an 1095  df-tru 1551  df-fal 1561  df-ex 1788  df-sb 2075  df-clab 2720  df-cleq 2733  df-clel 2816  df-ral 3056  df-rex 3066  df-rab 3394  df-v 3435  df-dif 3887  df-un 3889  df-in 3891  df-ss 3901  df-nul 4264  df-if 4457  df-sn 4558  df-pr 4560  df-op 4564  df-br 5075  df-opab 5137  df-id 5515  df-po 5528  df-xp 5626  df-rel 5627  df-res 5632

This theorem is referenced by: (None)