Theorem intirr 5091

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 3376	. . . 4 ⊢ (𝑅 ∩ I ) = ( I ∩ 𝑅)
2	1	eqeq1i 2217	. . 3 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ( I ∩ 𝑅) = ∅)
3		disj2 3527	. . 3 ⊢ (( I ∩ 𝑅) = ∅ ↔ I ⊆ (V ∖ 𝑅))
4		reli 4828	. . . 4 ⊢ Rel I
5		ssrel 4784	. . . 4 ⊢ (Rel I → ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))))
6	4, 5	ax-mp 5	. . 3 ⊢ ( I ⊆ (V ∖ 𝑅) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
7	2, 3, 6	3bitri 206	. 2 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
8		equcom 1732	. . . . 5 ⊢ (𝑦 = 𝑥 ↔ 𝑥 = 𝑦)
9		vex 2782	. . . . . 6 ⊢ 𝑦 ∈ V
10	9	ideq 4851	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
11		df-br 4063	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
12	8, 10, 11	3bitr2i 208	. . . 4 ⊢ (𝑦 = 𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
13		vex 2782	. . . . . . . 8 ⊢ 𝑥 ∈ V
14	13, 9	opex 4294	. . . . . . 7 ⊢ ⟨𝑥, 𝑦⟩ ∈ V
15	14	biantrur 303	. . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
16		eldif 3186	. . . . . 6 ⊢ (⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ V ∧ ¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅))
17	15, 16	bitr4i 187	. . . . 5 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝑅 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
18		df-br 4063	. . . . 5 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
19	17, 18	xchnxbir 685	. . . 4 ⊢ (¬ 𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅))
20	12, 19	imbi12i 239	. . 3 ⊢ ((𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ (⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
21	20	2albii 1497	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ I → ⟨𝑥, 𝑦⟩ ∈ (V ∖ 𝑅)))
22		nfv 1554	. . . 4 ⊢ Ⅎ𝑦 ¬ 𝑥𝑅𝑥
23		breq2 4066	. . . . 5 ⊢ (𝑦 = 𝑥 → (𝑥𝑅𝑦 ↔ 𝑥𝑅𝑥))
24	23	notbid 671	. . . 4 ⊢ (𝑦 = 𝑥 → (¬ 𝑥𝑅𝑦 ↔ ¬ 𝑥𝑅𝑥))
25	22, 24	equsal 1753	. . 3 ⊢ (∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ¬ 𝑥𝑅𝑥)
26	25	albii 1496	. 2 ⊢ (∀𝑥∀𝑦(𝑦 = 𝑥 → ¬ 𝑥𝑅𝑦) ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)
27	7, 21, 26	3bitr2i 208	1 ⊢ ((𝑅 ∩ I ) = ∅ ↔ ∀𝑥 ¬ 𝑥𝑅𝑥)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1373   = wceq 1375   ∈ wcel 2180  Vcvv 2779   ∖ cdif 3174   ∩ cin 3176   ⊆ wss 3177  ∅c0 3471  ⟨cop 3649   class class class wbr 4062   I cid 4356  Rel wrel 4701

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 617  ax-in2 618  ax-io 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272

This theorem depends on definitions:  df-bi 117  df-3an 985  df-tru 1378  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ral 2493  df-rex 2494  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-nul 3472  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-br 4063  df-opab 4125  df-id 4361  df-xp 4702  df-rel 4703

This theorem is referenced by: (None)