Theorem intirr 4731

Description: Two ways of saying a relation is irreflexive. Definition of irreflexivity in [Schechter] p. 51. (Contributed by NM, 9-Sep-2004.) (Proof shortened by Andrew Salmon, 27-Aug-2011.)

Assertion

Ref	Expression
intirr	⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)

Distinct variable group:   𝑥,𝑅

Proof of Theorem intirr

Dummy variable 𝑦 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		incom 3158	. . . 4 ⊢ (𝑅 ∩ I ) = ( I ∩ 𝑅)
2	1	eqeq1i 2088	. . 3 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3		disj2 3299	. . 3 ⊢ (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4		reli 4483	. . . 4 ⊢ Rel I
5		ssrel 4446	. . . 4 ⊢ (Rel I → ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))))
6	4, 5	ax-mp 7	. . 3 ⊢ ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
7	2, 3, 6	3bitri 204	. 2 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
8		equcom 1633	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
9		vex 2604	. . . . . 6 ⊢ 𝑦 ∈ V
10	9	ideq 4506	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
11		df-br 3786	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
12	8, 10, 11	3bitr2i 206	. . . 4 ⊢ (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13		vex 2604	. . . . . . . 8 ⊢ 𝑥 ∈ V
14	13, 9	opex 3984	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
15	14	biantrur 297	. . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
16		eldif 2982	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17	15, 16	bitr4i 185	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
18		df-br 3786	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
19	17, 18	xchnxbir 638	. . . 4 ⊢ (¬ 𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
20	12, 19	imbi12i 237	. . 3 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
21	20	2albii 1400	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
22		nfv 1461	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝑥𝑅𝑥
23		breq2 3789	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥))
24	23	notbid 624	. . . 4 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
25	22, 24	equsal 1655	. . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
26	25	albii 1399	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
27	7, 21, 26	3bitr2i 206	1 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 102   ↔ wb 103  ∀wal 1282   = wceq 1284   ∈ wcel 1433  Vcvv 2601   ∖ cdif 2970   ∩ cin 2972   ⊆ wss 2973  ∅c0 3251  ⟨cop 3401   class class class wbr 3785   I cid 4043  Rel wrel 4368

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 576  ax-in2 577  ax-io 662  ax-5 1376  ax-7 1377  ax-gen 1378  ax-ie1 1422  ax-ie2 1423  ax-8 1435  ax-10 1436  ax-11 1437  ax-i12 1438  ax-bndl 1439  ax-4 1440  ax-14 1445  ax-17 1459  ax-i9 1463  ax-ial 1467  ax-i5r 1468  ax-ext 2063  ax-sep 3896  ax-pow 3948  ax-pr 3964

This theorem depends on definitions:  df-bi 115  df-3an 921  df-tru 1287  df-nf 1390  df-sb 1686  df-eu 1944  df-mo 1945  df-clab 2068  df-cleq 2074  df-clel 2077  df-nfc 2208  df-ral 2353  df-rex 2354  df-v 2603  df-dif 2975  df-un 2977  df-in 2979  df-ss 2986  df-nul 3252  df-pw 3384  df-sn 3404  df-pr 3405  df-op 3407  df-br 3786  df-opab 3840  df-id 4048  df-xp 4369  df-rel 4370

This theorem is referenced by: (None)