Theorem intasym 4918

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

Assertion

Ref	Expression
intasym	⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))

Distinct variable group:   𝑥,𝑦,𝑅

Proof of Theorem intasym

Step	Hyp	Ref	Expression
1		relcnv 4912	. . 3 ⊢ Rel ◡𝑅
2		relin2 4653	. . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
3		ssrel 4622	. . 3 ⊢ (Rel (𝑅 ∩ ◡𝑅) → ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I )))
4	1, 2, 3	mp2b 8	. 2 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ))
5		elin 3254	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
6		df-br 3925	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7		vex 2684	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 2684	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 4717	. . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		df-br 3925	. . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
11	9, 10	bitr3i 185	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
12	6, 11	anbi12i 455	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
13	5, 12	bitr4i 186	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
14		df-br 3925	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
15	8	ideq 4686	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
16	14, 15	bitr3i 185	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
17	13, 16	imbi12i 238	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
18	17	2albii 1447	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
19	4, 18	bitri 183	1 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 103   ↔ wb 104  ∀wal 1329   ∈ wcel 1480   ∩ cin 3065   ⊆ wss 3066  ⟨cop 3525   class class class wbr 3924   I cid 4205  ^◡ccnv 4533  Rel wrel 4539

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ral 2419  df-rex 2420  df-v 2683  df-un 3070  df-in 3072  df-ss 3079  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-br 3925  df-opab 3985  df-id 4210  df-xp 4540  df-rel 4541  df-cnv 4542

This theorem is referenced by: (None)