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Theorem reldm0 5771
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 5645 . . 3 Rel ∅
2 eqrel 5631 . . 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 4281 . . 3 (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴)
5 alnex 1783 . . . . . 6 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
6 vex 3474 . . . . . . 7 𝑥 ∈ V
76eldm2 5743 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
85, 7xchbinxr 338 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴)
9 noel 4270 . . . . . . 7 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
109nbn 376 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1110albii 1821 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
128, 11bitr3i 280 . . . 4 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1312albii 1821 . . 3 (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
144, 13bitr2i 279 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ dom 𝐴 = ∅)
153, 14syl6bb 290 1 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 209  wal 1536   = wceq 1538  wex 1781  wcel 2115  c0 4266  cop 4546  dom cdm 5528  Rel wrel 5533
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 1971  ax-7 2016  ax-8 2117  ax-9 2125  ax-10 2146  ax-11 2162  ax-12 2178  ax-ext 2793
This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3an 1086  df-tru 1541  df-ex 1782  df-nf 1786  df-sb 2071  df-clab 2800  df-cleq 2814  df-clel 2892  df-nfc 2960  df-rab 3135  df-v 3473  df-dif 3913  df-un 3915  df-in 3917  df-ss 3927  df-nul 4267  df-if 4441  df-sn 4541  df-pr 4543  df-op 4547  df-br 5040  df-opab 5102  df-xp 5534  df-rel 5535  df-dm 5538
This theorem is referenced by:  relrn0  5813  coeq0  6081  fnresdisj  6440  fn0  6452  fresaunres2  6523  funopsn  6883  fsnunfv  6922  frxp  7795  domss2  8652  swrd0  13999  setsres  16504  pmtrsn  18626  gsumval3  19006  00lsp  19729  metn0  22946  wlkn0  27389  eulerpath  28005  funresdm1  30342  dfrdg2  33048  mbfresfi  34989  mapfzcons1  39469  diophrw  39511  eldioph2lem1  39512  eldioph2lem2  39513  sge0cl  42843
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