Theorem reldm0 2550

Description: A relation is empty iff its domain is empty.

Assertion

Ref	Expression
reldm0	⊢ (Rel A → (A = ∅ ↔ dom A = ∅))

Proof of Theorem reldm0

Step	Hyp	Ref	Expression
1		rel0 2499	. . 3 ⊢ Rel ∅
2		cleqrel 2483	. . 3 ⊢ ((Rel A ∧ Rel ∅) → (A = ∅ ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅)))
3	1, 2	mpan2 519	. 2 ⊢ (Rel A → (A = ∅ ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅)))
4		eq0 1719	. . 3 ⊢ (dom A = ∅ ↔ ∀x ¬ x ∈ dom A)
5		visset 1350	. . . . . . 7 ⊢ x ∈ V
6	5	eldm2 2528	. . . . . 6 ⊢ (x ∈ dom A ↔ ∃y⟨x, y⟩ ∈ A)
7	6	negbii 162	. . . . 5 ⊢ (¬ x ∈ dom A ↔ ¬ ∃y⟨x, y⟩ ∈ A)
8		alnex 716	. . . . . 6 ⊢ (∀y ¬ ⟨x, y⟩ ∈ A ↔ ¬ ∃y⟨x, y⟩ ∈ A)
9		noel 1711	. . . . . . . 8 ⊢ ¬ ⟨x, y⟩ ∈ ∅
10	9	nbn 542	. . . . . . 7 ⊢ (¬ ⟨x, y⟩ ∈ A ↔ (⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅))
11	10	bial 695	. . . . . 6 ⊢ (∀y ¬ ⟨x, y⟩ ∈ A ↔ ∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅))
12	8, 11	bitr3 153	. . . . 5 ⊢ (¬ ∃y⟨x, y⟩ ∈ A ↔ ∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅))
13	7, 12	bitr 151	. . . 4 ⊢ (¬ x ∈ dom A ↔ ∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅))
14	13	bial 695	. . 3 ⊢ (∀x ¬ x ∈ dom A ↔ ∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅))
15	4, 14	bitr2 152	. 2 ⊢ (∀x∀y(⟨x, y⟩ ∈ A ↔ ⟨x, y⟩ ∈ ∅) ↔ dom A = ∅)
16	3, 15	syl6bb 414	1 ⊢ (Rel A → (A = ∅ ↔ dom A = ∅))

Syntax hints:  ¬ wn 1   → wi 2   ↔ wb 127  ∀wal 672  ∃wex 678   = wceq 1091   ∈ wcel 1092  ∅c0 1707  ⟨cop 1810  dom cdm 2410  Rel wrel 2415

This theorem is referenced by:  relrn0 2568  fnresdisj 2732  mapdom2lem 3388

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-xp 2424  df-rel 2425  df-dm 2428

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