Theorem poirr2 5160

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.

Step	Hyp	Ref	Expression
1		relres 5071	. . . 4 ⊢ Rel ( I ↾ 𝐴)
2		relin2 4876	. . . 4 ⊢ (Rel ( I ↾ 𝐴) → Rel (𝑅 ∩ ( I ↾ 𝐴)))
3	1, 2	mp1i 10	. . 3 ⊢ (𝑅 Po 𝐴 → Rel (𝑅 ∩ ( I ↾ 𝐴)))
4		df-br 4115	. . . . 5 ⊢ (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)))
5		brin 4167	. . . . 5 ⊢ (𝑥(𝑅 ∩ ( I ↾ 𝐴))𝑦 ↔ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
6	4, 5	bitr3i 186	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) ↔ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
7		vex 2818	. . . . . . . . 9 ⊢ 𝑦 ∈ V
8	7	brres 5049	. . . . . . . 8 ⊢ (𝑥( I ↾ 𝐴)𝑦 ↔ (𝑥 I 𝑦 ∧ 𝑥 ∈ 𝐴))
9		poirr 4433	. . . . . . . . . . 11 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑥)
10	7	ideq 4912	. . . . . . . . . . . . 13 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
11		breq2 4118	. . . . . . . . . . . . 13 ⊢ (𝑥 = 𝑦 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑦))
12	10, 11	sylbi 121	. . . . . . . . . . . 12 ⊢ (𝑥 I 𝑦 → (𝑥𝑅𝑥 ↔ 𝑥𝑅𝑦))
13	12	notbid 673	. . . . . . . . . . 11 ⊢ (𝑥 I 𝑦 → (¬ 𝑥𝑅𝑥 ↔ ¬ 𝑥𝑅𝑦))
14	9, 13	syl5ibcom 155	. . . . . . . . . 10 ⊢ ((𝑅 Po 𝐴 ∧ 𝑥 ∈ 𝐴) → (𝑥 I 𝑦 → ¬ 𝑥𝑅𝑦))
15	14	expimpd 363	. . . . . . . . 9 ⊢ (𝑅 Po 𝐴 → ((𝑥 ∈ 𝐴 ∧ 𝑥 I 𝑦) → ¬ 𝑥𝑅𝑦))
16	15	ancomsd 269	. . . . . . . 8 ⊢ (𝑅 Po 𝐴 → ((𝑥 I 𝑦 ∧ 𝑥 ∈ 𝐴) → ¬ 𝑥𝑅𝑦))
17	8, 16	biimtrid 152	. . . . . . 7 ⊢ (𝑅 Po 𝐴 → (𝑥( I ↾ 𝐴)𝑦 → ¬ 𝑥𝑅𝑦))
18	17	con2d 629	. . . . . 6 ⊢ (𝑅 Po 𝐴 → (𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦))
19		imnan 697	. . . . . 6 ⊢ ((𝑥𝑅𝑦 → ¬ 𝑥( I ↾ 𝐴)𝑦) ↔ ¬ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
20	18, 19	sylib 122	. . . . 5 ⊢ (𝑅 Po 𝐴 → ¬ (𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦))
21	20	pm2.21d 624	. . . 4 ⊢ (𝑅 Po 𝐴 → ((𝑥𝑅𝑦 ∧ 𝑥( I ↾ 𝐴)𝑦) → ⟨𝑥, 𝑦⟩ ∈ ∅))
22	6, 21	biimtrid 152	. . 3 ⊢ (𝑅 Po 𝐴 → (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ( I ↾ 𝐴)) → ⟨𝑥, 𝑦⟩ ∈ ∅))
23	3, 22	relssdv 4847	. 2 ⊢ (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅)
24		ss0 3553	. 2 ⊢ ((𝑅 ∩ ( I ↾ 𝐴)) ⊆ ∅ → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)
25	23, 24	syl 14	1 ⊢ (𝑅 Po 𝐴 → (𝑅 ∩ ( I ↾ 𝐴)) = ∅)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105   = wceq 1398   ∈ wcel 2205   ∩ cin 3213   ⊆ wss 3214  ∅c0 3512  ⟨cop 3697   class class class wbr 4114   I cid 4414   Po wpo 4420   ↾ cres 4756  Rel wrel 4759

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327

This theorem depends on definitions:  df-bi 117  df-3an 1007  df-tru 1401  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ral 2527  df-rex 2528  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-nul 3513  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-br 4115  df-opab 4177  df-id 4419  df-po 4422  df-xp 4760  df-rel 4761  df-res 4766

This theorem is referenced by: (None)