Theorem intasym 5050

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 5043	. . 3 ⊢ Rel ◡𝑅
2		relin2 4778	. . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
3		ssrel 4747	. . 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 3342	. . . . 5 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
6		df-br 4030	. . . . . 6 ⊢ (𝑥𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ 𝑅)
7		vex 2763	. . . . . . . 8 ⊢ 𝑥 ∈ V
8		vex 2763	. . . . . . . 8 ⊢ 𝑦 ∈ V
9	7, 8	brcnv 4845	. . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ 𝑦𝑅𝑥)
10		df-br 4030	. . . . . . 7 ⊢ (𝑥◡𝑅𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
11	9, 10	bitr3i 186	. . . . . 6 ⊢ (𝑦𝑅𝑥 ↔ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅)
12	6, 11	anbi12i 460	. . . . 5 ⊢ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) ↔ (⟨𝑥, 𝑦⟩ ∈ 𝑅 ∧ ⟨𝑥, 𝑦⟩ ∈ ◡𝑅))
13	5, 12	bitr4i 187	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥))
14		df-br 4030	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ ⟨𝑥, 𝑦⟩ ∈ I )
15	8	ideq 4814	. . . . 5 ⊢ (𝑥 I 𝑦 ↔ 𝑥 = 𝑦)
16	14, 15	bitr3i 186	. . . 4 ⊢ (⟨𝑥, 𝑦⟩ ∈ I ↔ 𝑥 = 𝑦)
17	13, 16	imbi12i 239	. . 3 ⊢ ((⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
18	17	2albii 1482	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨𝑥, 𝑦⟩ ∈ I ) ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))
19	4, 18	bitri 184	1 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀𝑥∀𝑦((𝑥𝑅𝑦 ∧ 𝑦𝑅𝑥) → 𝑥 = 𝑦))

Syntax hints:   → wi 4   ∧ wa 104   ↔ wb 105  ∀wal 1362   ∈ wcel 2164   ∩ cin 3152   ⊆ wss 3153  ⟨cop 3621   class class class wbr 4029   I cid 4319  ^◡ccnv 4658  Rel wrel 4664

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-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-id 4324  df-xp 4665  df-rel 4666  df-cnv 4667

This theorem is referenced by: (None)