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Theorem intasym 6105

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 6096	. . 3 ⊢ Rel ^◡𝑅
2		relin2 5790	. . 3 ⊢ (Rel ^◡𝑅 → Rel (𝑅 ∩ ^◡𝑅))
3		ssrel 5759	. . 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 3923	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅))
6		df-br 5105	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7		vex 3461	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 3461	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 5858	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		df-br 5105	. . . . . . 7 ⊢ (𝑥^◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅)
11	9, 10	bitr3i 280	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅)
12	6, 11	anbi12i 639	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ^◡𝑅))
13	5, 12	bitr4i 281	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
14		df-br 5105	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
15	8	ideq 5828	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
16	14, 15	bitr3i 280	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
17	13, 16	imbi12i 353	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
18	17	2albii 1843	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ^◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
19	4, 18	bitri 278	1 ⊢ ((𝑅 ∩ ^◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 400 ∀wal 1561 ∈ wcel 2145 ∩ cin 3906 ⊆ wss 3907 ⟨cop 4591 class class class wbr 5104 I cid 5545 ^◡ccnv 5650 Rel wrel 5656

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1818 ax-4 1832 ax-5 1933 ax-6 1990 ax-7 2031 ax-8 2147 ax-9 2155 ax-ext 2737 ax-sep 5250 ax-pr 5394

This theorem depends on definitions: df-bi 210 df-an 401 df-or 861 df-3an 1103 df-tru 1566 df-fal 1576 df-ex 1803 df-sb 2094 df-clab 2744 df-cleq 2757 df-clel 2840 df-ral 3080 df-rex 3090 df-rab 3418 df-v 3459 df-dif 3910 df-un 3912 df-in 3914 df-ss 3924 df-nul 4289 df-if 4484 df-sn 4586 df-pr 4588 df-op 4592 df-br 5105 df-opab 5167 df-id 5546 df-xp 5657 df-rel 5658 df-cnv 5659

This theorem is referenced by: (None)

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