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Theorem reldm0 5928
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.
StepHypRef Expression
1 rel0 5800 . . 3 Rel ∅
2 eqrel 5785 . . 3 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
31, 2mpan2 690 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
4 eq0 4344 . . 3 (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴)
5 alnex 1784 . . . . . 6 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
6 vex 3479 . . . . . . 7 𝑥 ∈ V
76eldm2 5902 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
85, 7xchbinxr 335 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴)
9 noel 4331 . . . . . . 7 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
109nbn 373 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1110albii 1822 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
128, 11bitr3i 277 . . . 4 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1312albii 1822 . . 3 (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
144, 13bitr2i 276 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ dom 𝐴 = ∅)
153, 14bitrdi 287 1 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 205  wal 1540   = wceq 1542  wex 1782  wcel 2107  c0 4323  cop 4635  dom cdm 5677  Rel wrel 5682
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704
This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-3an 1090  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-rab 3434  df-v 3477  df-dif 3952  df-un 3954  df-in 3956  df-ss 3966  df-nul 4324  df-if 4530  df-sn 4630  df-pr 4632  df-op 4636  df-br 5150  df-opab 5212  df-xp 5683  df-rel 5684  df-dm 5687
This theorem is referenced by:  relrn0  5969  relresdm1  6034  coeq0  6255  snres0  6298  fnresdisj  6671  fn0  6682  fresaunres2  6764  funopsn  7146  fsnunfv  7185  frxp  8112  frxp2  8130  frxp3  8137  domss2  9136  swrd0  14608  setsres  17111  pmtrsn  19387  gsumval3  19775  00lsp  20592  metn0  23866  noetasuplem2  27237  noetainflem2  27241  wlkn0  28878  eulerpath  29494  dfrdg2  34767  mbfresfi  36534  mapfzcons1  41455  diophrw  41497  eldioph2lem1  41498  eldioph2lem2  41499  tfsconcatb0  42094  tfsconcat0i  42095  tfsconcat0b  42096  sge0cl  45097
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