Theorem intasym 4652

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 4646	. . 3 ⊢ Rel ◡𝑅
2		relin2 4399	. . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
3		ssrel 4371	. . 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 3120	. . . . 5 ⊢ (⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨x, y⟩ ∈ 𝑅 ∧ ⟨x, y⟩ ∈ ◡𝑅))
6		df-br 3756	. . . . . 6 ⊢ (x𝑅y ↔ ⟨x, y⟩ ∈ 𝑅)
7		vex 2554	. . . . . . . 8 ⊢ x ∈ V
8		vex 2554	. . . . . . . 8 ⊢ y ∈ V
9	7, 8	brcnv 4461	. . . . . . 7 ⊢ (x◡𝑅y ↔ y𝑅x)
10		df-br 3756	. . . . . . 7 ⊢ (x◡𝑅y ↔ ⟨x, y⟩ ∈ ◡𝑅)
11	9, 10	bitr3i 175	. . . . . 6 ⊢ (y𝑅x ↔ ⟨x, y⟩ ∈ ◡𝑅)
12	6, 11	anbi12i 433	. . . . 5 ⊢ ((x𝑅y ∧ y𝑅x) ↔ (⟨x, y⟩ ∈ 𝑅 ∧ ⟨x, y⟩ ∈ ◡𝑅))
13	5, 12	bitr4i 176	. . . 4 ⊢ (⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (x𝑅y ∧ y𝑅x))
14		df-br 3756	. . . . 5 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
15	8	ideq 4431	. . . . 5 ⊢ (x I y ↔ x = y)
16	14, 15	bitr3i 175	. . . 4 ⊢ (⟨x, y⟩ ∈ I ↔ x = y)
17	13, 16	imbi12i 228	. . 3 ⊢ ((⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨x, y⟩ ∈ I ) ↔ ((x𝑅y ∧ y𝑅x) → x = y))
18	17	2albii 1357	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) → ⟨x, y⟩ ∈ I ) ↔ ∀x∀y((x𝑅y ∧ y𝑅x) → x = y))
19	4, 18	bitri 173	1 ⊢ ((𝑅 ∩ ◡𝑅) ⊆ I ↔ ∀x∀y((x𝑅y ∧ y𝑅x) → x = y))

Syntax hints:   → wi 4   ∧ wa 97   ↔ wb 98  ∀wal 1240   ∈ wcel 1390   ∩ cin 2910   ⊆ wss 2911  ⟨cop 3370   class class class wbr 3755   I cid 4016  ^◡ccnv 4287  Rel wrel 4293

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bnd 1396  ax-4 1397  ax-14 1402  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019  ax-sep 3866  ax-pow 3918  ax-pr 3935

This theorem depends on definitions:  df-bi 110  df-3an 886  df-tru 1245  df-nf 1347  df-sb 1643  df-eu 1900  df-mo 1901  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-ral 2305  df-rex 2306  df-v 2553  df-un 2916  df-in 2918  df-ss 2925  df-pw 3353  df-sn 3373  df-pr 3374  df-op 3376  df-br 3756  df-opab 3810  df-id 4021  df-xp 4294  df-rel 4295  df-cnv 4296

This theorem is referenced by: (None)