Theorem intasym 4625

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 4619	. . 3 ⊢ Rel ◡𝑅
2		relin2 4372	. . 3 ⊢ (Rel ◡𝑅 → Rel (𝑅 ∩ ◡𝑅))
3		ssrel 4344	. . 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 3095	. . . . 5 ⊢ (⟨x, y⟩ ∈ (𝑅 ∩ ◡𝑅) ↔ (⟨x, y⟩ ∈ 𝑅 ∧ ⟨x, y⟩ ∈ ◡𝑅))
6		df-br 3729	. . . . . 6 ⊢ (x𝑅y ↔ ⟨x, y⟩ ∈ 𝑅)
7		vex 2530	. . . . . . . 8 ⊢ x ∈ V
8		vex 2530	. . . . . . . 8 ⊢ y ∈ V
9	7, 8	brcnv 4434	. . . . . . 7 ⊢ (x◡𝑅y ↔ y𝑅x)
10		df-br 3729	. . . . . . 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 3729	. . . . 5 ⊢ (x I y ↔ ⟨x, y⟩ ∈ I )
15	8	ideq 4404	. . . . 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 1334	. 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 1222   ∈ wcel 1367   ∩ cin 2885   ⊆ wss 2886  ⟨cop 3343   class class class wbr 3728   I cid 3989  ^◡ccnv 4260  Rel wrel 4266

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 614  ax-5 1310  ax-7 1311  ax-gen 1312  ax-ie1 1356  ax-ie2 1357  ax-8 1369  ax-10 1370  ax-11 1371  ax-i12 1372  ax-bnd 1373  ax-4 1374  ax-14 1379  ax-17 1393  ax-i9 1397  ax-ial 1401  ax-i5r 1402  ax-ext 1996  ax-sep 3839  ax-pow 3891  ax-pr 3908

This theorem depends on definitions:  df-bi 110  df-3an 869  df-tru 1227  df-nf 1324  df-sb 1620  df-eu 1877  df-mo 1878  df-clab 2001  df-cleq 2007  df-clel 2010  df-nfc 2141  df-ral 2281  df-rex 2282  df-v 2529  df-un 2891  df-in 2893  df-ss 2900  df-pw 3326  df-sn 3346  df-pr 3347  df-op 3349  df-br 3729  df-opab 3783  df-id 3994  df-xp 4267  df-rel 4268  df-cnv 4269

This theorem is referenced by: (None)