intasym - Metamath Proof Explorer

	Metamath Proof Explorer		< Previous Next > Nearby theorems
Mirrors > Home > MPE Home > Th. List > intasym		Structured version Visualization version GIF version

Theorem intasym 6113

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 6100	. . 3 ⊢ Rel ^◡𝑅
2		relin2 5811	. . 3 ⊢ (Rel ^◡𝑅 → Rel (𝑅 ∩ ^◡𝑅))
3		ssrel 5780	. . 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 3963	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅))
6		df-br 5148	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7		vex 3478	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 3478	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 5880	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		df-br 5148	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅)
11	9, 10	bitr3i 276	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅)
12	6, 11	anbi12i 627	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅))
13	5, 12	bitr4i 277	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
14		df-br 5148	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
15	8	ideq 5850	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
16	14, 15	bitr3i 276	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
17	13, 16	imbi12i 350	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
18	17	2albii 1822	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
19	4, 18	bitri 274	1 ⊢ ((𝑅 ∩ ^◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 396 ∀wal 1539 ∈ wcel 2106 ∩ cin 3946 ⊆ wss 3947 ⟨cop 4633 class class class wbr 5147 I cid 5572 ^◡ccnv 5674 Rel wrel 5680

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2703 ax-sep 5298 ax-nul 5305 ax-pr 5426

This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3an 1089 df-tru 1544 df-fal 1554 df-ex 1782 df-sb 2068 df-clab 2710 df-cleq 2724 df-clel 2810 df-ral 3062 df-rex 3071 df-rab 3433 df-v 3476 df-dif 3950 df-un 3952 df-in 3954 df-ss 3964 df-nul 4322 df-if 4528 df-sn 4628 df-pr 4630 df-op 4634 df-br 5148 df-opab 5210 df-id 5573 df-xp 5681 df-rel 5682 df-cnv 5683

This theorem is referenced by: (None)

W3C validator