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Theorem reldm0 5940
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 5811 . . 3 Rel ∅
2 eqrel 5796 . . 3 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
31, 2mpan2 691 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
4 eq0 4355 . . 3 (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴)
5 alnex 1777 . . . . . 6 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
6 vex 3481 . . . . . . 7 𝑥 ∈ V
76eldm2 5914 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
85, 7xchbinxr 335 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴)
9 noel 4343 . . . . . . 7 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
109nbn 372 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1110albii 1815 . . . . 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 1815 . . 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 206  wal 1534   = wceq 1536  wex 1775  wcel 2105  c0 4338  cop 4636  dom cdm 5688  Rel wrel 5693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1791  ax-4 1805  ax-5 1907  ax-6 1964  ax-7 2004  ax-8 2107  ax-9 2115  ax-ext 2705
This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1539  df-fal 1549  df-ex 1776  df-sb 2062  df-clab 2712  df-cleq 2726  df-clel 2813  df-rab 3433  df-v 3479  df-dif 3965  df-un 3967  df-ss 3979  df-nul 4339  df-if 4531  df-sn 4631  df-pr 4633  df-op 4637  df-br 5148  df-opab 5210  df-xp 5694  df-rel 5695  df-dm 5698
This theorem is referenced by:  relrn0  5985  relresdm1  6052  coeq0  6276  snres0  6319  fnresdisj  6688  fn0  6699  fresaunres2  6780  funopsn  7167  fsnunfv  7206  frxp  8149  frxp2  8167  frxp3  8174  domss2  9174  swrd0  14692  setsres  17211  pmtrsn  19551  gsumval3  19939  00lsp  20996  metn0  24385  noetasuplem2  27793  noetainflem2  27797  wlkn0  29653  eulerpath  30269  dfrdg2  35776  mbfresfi  37652  mapfzcons1  42704  diophrw  42746  eldioph2lem1  42747  eldioph2lem2  42748  tfsconcatb0  43333  tfsconcat0i  43334  tfsconcat0b  43335  sge0cl  46336
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