Theorem intirr 2628

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

Assertion

Ref	Expression
intirr	⊢ ((R ∩ I) = ∅ ↔ ∀x ¬ xRx)

Distinct variable group(s):   x,R

Proof of Theorem intirr

Step	Hyp	Ref	Expression
1		inss2 1658	. . . 4 ⊢ (R ∩ I) ⊆ I
2		reli 2500	. . . 4 ⊢ Rel I
3		ssrel 2479	. . . 4 ⊢ ((R ∩ I) ⊆ I → (Rel I → Rel (R ∩ I)))
4	1, 2, 3	mp2 43	. . 3 ⊢ Rel (R ∩ I)
5		rel0 2499	. . 3 ⊢ Rel ∅
6		cleqrel 2483	. . 3 ⊢ ((Rel (R ∩ I) ∧ Rel ∅) → ((R ∩ I) = ∅ ↔ ∀x∀y(⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅)))
7	4, 5, 6	mp2an 520	. 2 ⊢ ((R ∩ I) = ∅ ↔ ∀x∀y(⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅))
8		df-br 2063	. . . . . 6 ⊢ (xRx ↔ ⟨x, x⟩ ∈ R)
9		visset 1350	. . . . . . 7 ⊢ x ∈ V
10		opeq2 1877	. . . . . . . 8 ⊢ (y = x → ⟨x, y⟩ = ⟨x, x⟩)
11	10	eleq1d 1155	. . . . . . 7 ⊢ (y = x → (⟨x, y⟩ ∈ R ↔ ⟨x, x⟩ ∈ R))
12	9, 11	ceqsexv 1371	. . . . . 6 ⊢ (∃y(y = x ∧ ⟨x, y⟩ ∈ R) ↔ ⟨x, x⟩ ∈ R)
13	8, 12	bitr4 154	. . . . 5 ⊢ (xRx ↔ ∃y(y = x ∧ ⟨x, y⟩ ∈ R))
14		noel 1711	. . . . . . . . 9 ⊢ ¬ ⟨x, y⟩ ∈ ∅
15	14	nbn 542	. . . . . . . 8 ⊢ (¬ ⟨x, y⟩ ∈ (R ∩ I) ↔ (⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅))
16	15	bicon1i 193	. . . . . . 7 ⊢ (¬ (⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅) ↔ ⟨x, y⟩ ∈ (R ∩ I))
17		visset 1350	. . . . . . . . . . 11 ⊢ y ∈ V
18	9, 17	ideq 2127	. . . . . . . . . 10 ⊢ (xIy ↔ x = y)
19		df-br 2063	. . . . . . . . . 10 ⊢ (xIy ↔ ⟨x, y⟩ ∈ I)
20		cleqcom 1103	. . . . . . . . . 10 ⊢ (x = y ↔ y = x)
21	18, 19, 20	3bitr3r 157	. . . . . . . . 9 ⊢ (y = x ↔ ⟨x, y⟩ ∈ I)
22	21	anbi2i 367	. . . . . . . 8 ⊢ ((⟨x, y⟩ ∈ R ∧ y = x) ↔ (⟨x, y⟩ ∈ R ∧ ⟨x, y⟩ ∈ I))
23		ancom 333	. . . . . . . 8 ⊢ ((y = x ∧ ⟨x, y⟩ ∈ R) ↔ (⟨x, y⟩ ∈ R ∧ y = x))
24		elin 1635	. . . . . . . 8 ⊢ (⟨x, y⟩ ∈ (R ∩ I) ↔ (⟨x, y⟩ ∈ R ∧ ⟨x, y⟩ ∈ I))
25	22, 23, 24	3bitr4r 159	. . . . . . 7 ⊢ (⟨x, y⟩ ∈ (R ∩ I) ↔ (y = x ∧ ⟨x, y⟩ ∈ R))
26	16, 25	bitr2 152	. . . . . 6 ⊢ ((y = x ∧ ⟨x, y⟩ ∈ R) ↔ ¬ (⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅))
27	26	biex 733	. . . . 5 ⊢ (∃y(y = x ∧ ⟨x, y⟩ ∈ R) ↔ ∃y ¬ (⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅))
28		exnal 721	. . . . 5 ⊢ (∃y ¬ (⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅) ↔ ¬ ∀y(⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅))
29	13, 27, 28	3bitr 155	. . . 4 ⊢ (xRx ↔ ¬ ∀y(⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅))
30	29	bicon2i 194	. . 3 ⊢ (∀y(⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅) ↔ ¬ xRx)
31	30	bial 695	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ (R ∩ I) ↔ ⟨x, y⟩ ∈ ∅) ↔ ∀x ¬ xRx)
32	7, 31	bitr 151	1 ⊢ ((R ∩ I) = ∅ ↔ ∀x ¬ xRx)

Syntax hints:  ¬ wn 1   ↔ wb 127   ∧ wa 196  ∀wal 672  ∃wex 678   = weq 797   = wceq 1091   ∈ wcel 1092   ∩ cin 1486   ⊆ wss 1487  ∅c0 1707  ⟨cop 1810   class class class wbr 2054  Icid 2057  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

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