Theorem intasym 2627

Description: Two ways of saying a relation is antisymmetric. Definition of antisymmetry in [Schechter] p. 51.

Assertion

Ref	Expression
intasym	⊢ ((R ∩ ◡R) ⊆ I ↔ ∀x∀y((xRy ∧ yRx) → x = y))

Distinct variable group(s):   x,y,R

Proof of Theorem intasym

Step	Hyp	Ref	Expression
1		inss2 1658	. . . 4 ⊢ (R ∩ ◡R) ⊆ ◡R
2		relcnv 2624	. . . 4 ⊢ Rel ◡R
3		ssrel 2479	. . . 4 ⊢ ((R ∩ ◡R) ⊆ ◡R → (Rel ◡R → Rel (R ∩ ◡R)))
4	1, 2, 3	mp2 43	. . 3 ⊢ Rel (R ∩ ◡R)
5		relss 2480	. . 3 ⊢ (Rel (R ∩ ◡R) → ((R ∩ ◡R) ⊆ I ↔ ∀x∀y(⟨x, y⟩ ∈ (R ∩ ◡R) → ⟨x, y⟩ ∈ I)))
6	4, 5	ax-mp 6	. 2 ⊢ ((R ∩ ◡R) ⊆ I ↔ ∀x∀y(⟨x, y⟩ ∈ (R ∩ ◡R) → ⟨x, y⟩ ∈ I))
7		df-br 2063	. . . . . 6 ⊢ (xRy ↔ ⟨x, y⟩ ∈ R)
8		visset 1350	. . . . . . . 8 ⊢ x ∈ V
9		visset 1350	. . . . . . . 8 ⊢ y ∈ V
10	8, 9	brcnv 2519	. . . . . . 7 ⊢ (x◡Ry ↔ yRx)
11		df-br 2063	. . . . . . 7 ⊢ (x◡Ry ↔ ⟨x, y⟩ ∈ ◡R)
12	10, 11	bitr3 153	. . . . . 6 ⊢ (yRx ↔ ⟨x, y⟩ ∈ ◡R)
13	7, 12	anbi12i 369	. . . . 5 ⊢ ((xRy ∧ yRx) ↔ (⟨x, y⟩ ∈ R ∧ ⟨x, y⟩ ∈ ◡R))
14		elin 1635	. . . . 5 ⊢ (⟨x, y⟩ ∈ (R ∩ ◡R) ↔ (⟨x, y⟩ ∈ R ∧ ⟨x, y⟩ ∈ ◡R))
15	13, 14	bitr4 154	. . . 4 ⊢ ((xRy ∧ yRx) ↔ ⟨x, y⟩ ∈ (R ∩ ◡R))
16	8, 9	ideq 2127	. . . . 5 ⊢ (xIy ↔ x = y)
17		df-br 2063	. . . . 5 ⊢ (xIy ↔ ⟨x, y⟩ ∈ I)
18	16, 17	bitr3 153	. . . 4 ⊢ (x = y ↔ ⟨x, y⟩ ∈ I)
19	15, 18	imbi12i 163	. . 3 ⊢ (((xRy ∧ yRx) → x = y) ↔ (⟨x, y⟩ ∈ (R ∩ ◡R) → ⟨x, y⟩ ∈ I))
20	19	bi2al 696	. 2 ⊢ (∀x∀y((xRy ∧ yRx) → x = y) ↔ ∀x∀y(⟨x, y⟩ ∈ (R ∩ ◡R) → ⟨x, y⟩ ∈ I))
21	6, 20	bitr4 154	1 ⊢ ((R ∩ ◡R) ⊆ I ↔ ∀x∀y((xRy ∧ yRx) → x = y))

Syntax hints:   → wi 2   ↔ wb 127   ∧ wa 196  ∀wal 672   = weq 797   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487  ⟨cop 1810   class class class wbr 2054  Icid 2057  ^◡ccnv 2409  Rel wrel 2415

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-13 804  ax-14 805  ax-16 922  ax-17 925  ax-ext 1074  ax-rep 1075  ax-pow 1077

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-un 1490  df-in 1491  df-ss 1492  df-nul 1708  df-pw 1799  df-sn 1811  df-pr 1812  df-op 1815  df-br 2063  df-opab 2098  df-id 2125  df-xp 2424  df-rel 2425  df-cnv 2426

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