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Theorem intasym 5511

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 5503	. . 3 ⊢ Rel ^◡𝑅
2		relin2 5237	. . 3 ⊢ (Rel ^◡𝑅 → Rel (𝑅 ∩ ^◡𝑅))
3		ssrel 5207	. . 3 ⊢ (Rel (𝑅 ∩ ^◡𝑅) → ((𝑅 ∩ ^◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I )))
4	1, 2, 3	mp2b 10	. 2 ⊢ ((𝑅 ∩ ^◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ))
5		elin 3796	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅))
6		df-br 4654	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7		vex 3203	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 3203	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 5305	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		df-br 4654	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅)
11	9, 10	bitr3i 266	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅)
12	6, 11	anbi12i 733	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅))
13	5, 12	bitr4i 267	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
14		df-br 4654	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
15	8	ideq 5274	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
16	14, 15	bitr3i 266	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
17	13, 16	imbi12i 340	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
18	17	2albii 1748	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
19	4, 18	bitri 264	1 ⊢ ((𝑅 ∩ ^◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 384 ∀wal 1481 ∈ wcel 1990 ∩ cin 3573 ⊆ wss 3574 ⟨cop 4183 class class class wbr 4653 I cid 5023 ^◡ccnv 5113 Rel wrel 5119

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 ax-sep 4781 ax-nul 4789 ax-pr 4906

This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3an 1039 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-eu 2474 df-mo 2475 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-ral 2917 df-rex 2918 df-rab 2921 df-v 3202 df-dif 3577 df-un 3579 df-in 3581 df-ss 3588 df-nul 3916 df-if 4087 df-sn 4178 df-pr 4180 df-op 4184 df-br 4654 df-opab 4713 df-id 5024 df-xp 5120 df-rel 5121 df-cnv 5122

This theorem is referenced by: (None)

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