Theorem reldm0 5883

Description: A relation is empty iff its domain is empty. (Contributed by NM, 15-Sep-2004.)

Assertion

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

Proof of Theorem reldm0

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		rel0 5755	. . 3 ⊢ Rel ∅
2		eqrel 5740	. . 3 ⊢ ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
3	1, 2	mpan2 692	. 2 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
4		eq0 4290	. . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴)
5		alnex 1783	. . . . . 6 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
6		vex 3433	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	eldm2 5856	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
8	5, 7	xchbinxr 335	. . . . 5 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴)
9		noel 4278	. . . . . . 7 ⊢ ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
10	9	nbn 372	. . . . . 6 ⊢ (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
11	10	albii 1821	. . . . 5 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
12	8, 11	bitr3i 277	. . . 4 ⊢ (¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
13	12	albii 1821	. . 3 ⊢ (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
14	4, 13	bitr2i 276	. 2 ⊢ (∀𝑥∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ dom 𝐴 = ∅)
15	3, 14	bitrdi 287	1 ⊢ (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206  ∀wal 1540   = wceq 1542  ∃wex 1781   ∈ wcel 2114  ∅c0 4273  ⟨cop 4573  dom cdm 5631  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-3an 1089  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-rab 3390  df-v 3431  df-dif 3892  df-un 3894  df-ss 3906  df-nul 4274  df-if 4467  df-sn 4568  df-pr 4570  df-op 4574  df-br 5086  df-opab 5148  df-xp 5637  df-rel 5638  df-dm 5641

This theorem is referenced by:  relrn0  5928  relresdm1  5998  coeq0  6220  snres0  6262  fnresdisj  6618  fn0  6629  fresaunres2  6712  funopsnOLD  7102  fsnunfv  7142  frxp  8076  frxp2  8094  frxp3  8101  domss2  9074  swrd0  14621  setsres  17148  pmtrsn  19494  gsumval3  19882  00lsp  20976  metn0  24325  noetasuplem2  27698  noetainflem2  27702  wlkn0  29689  eulerpath  30311  dfrdg2  35975  mbfresfi  37987  mapfzcons1  43149  diophrw  43191  eldioph2lem1  43192  eldioph2lem2  43193  tfsconcatb0  43772  tfsconcat0i  43773  tfsconcat0b  43774  sge0cl  46809  resinsn  49347  resinsnALT  49348