Theorem reldm0 5877

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 5748	. . 3 ⊢ Rel ∅
2		eqrel 5733	. . 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 4291	. . 3 ⊢ (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴)
5		alnex 1783	. . . . . 6 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
6		vex 3434	. . . . . . 7 ⊢ 𝑥 ∈ V
7	6	eldm2 5850	. . . . . 6 ⊢ (𝑥 ∈ dom 𝐴 ↔ ∃𝑦⟨𝑥, 𝑦⟩ ∈ 𝐴)
8	5, 7	xchbinxr 335	. . . . 5 ⊢ (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴)
9		noel 4279	. . . . . . 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 4274  ⟨cop 4574  dom cdm 5624  Rel wrel 5629

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 2709

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 2716  df-cleq 2729  df-clel 2812  df-rab 3391  df-v 3432  df-dif 3893  df-un 3895  df-ss 3907  df-nul 4275  df-if 4468  df-sn 4569  df-pr 4571  df-op 4575  df-br 5087  df-opab 5149  df-xp 5630  df-rel 5631  df-dm 5634

This theorem is referenced by:  relrn0  5922  relresdm1  5992  coeq0  6214  snres0  6256  fnresdisj  6612  fn0  6623  fresaunres2  6706  funopsn  7095  fsnunfv  7135  frxp  8069  frxp2  8087  frxp3  8094  domss2  9067  swrd0  14612  setsres  17139  pmtrsn  19485  gsumval3  19873  00lsp  20967  metn0  24335  noetasuplem2  27712  noetainflem2  27716  wlkn0  29704  eulerpath  30326  dfrdg2  35991  mbfresfi  38001  mapfzcons1  43163  diophrw  43205  eldioph2lem1  43206  eldioph2lem2  43207  tfsconcatb0  43790  tfsconcat0i  43791  tfsconcat0b  43792  sge0cl  46827  resinsn  49359  resinsnALT  49360