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Theorem reldm0 5480
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 5381 . . 3 Rel ∅
2 eqrel 5348 . . 3 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
31, 2mpan2 671 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
4 eq0 4077 . . 3 (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴)
5 alnex 1854 . . . . . 6 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
6 vex 3354 . . . . . . 7 𝑥 ∈ V
76eldm2 5459 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
85, 7xchbinxr 324 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴)
9 noel 4067 . . . . . . 7 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
109nbn 361 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1110albii 1895 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
128, 11bitr3i 266 . . . 4 𝑥 ∈ dom 𝐴 ↔ ∀𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1312albii 1895 . . 3 (∀𝑥 ¬ 𝑥 ∈ dom 𝐴 ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
144, 13bitr2i 265 . 2 (∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅) ↔ dom 𝐴 = ∅)
153, 14syl6bb 276 1 (Rel 𝐴 → (𝐴 = ∅ ↔ dom 𝐴 = ∅))
Colors of variables: wff setvar class
Syntax hints:  ¬ wn 3  wi 4  wb 196  wal 1629   = wceq 1631  wex 1852  wcel 2145  c0 4063  cop 4323  dom cdm 5250  Rel wrel 5255
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1870  ax-4 1885  ax-5 1991  ax-6 2057  ax-7 2093  ax-9 2154  ax-10 2174  ax-11 2190  ax-12 2203  ax-13 2408  ax-ext 2751
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 837  df-3an 1073  df-tru 1634  df-ex 1853  df-nf 1858  df-sb 2050  df-clab 2758  df-cleq 2764  df-clel 2767  df-nfc 2902  df-rab 3070  df-v 3353  df-dif 3726  df-un 3728  df-in 3730  df-ss 3737  df-nul 4064  df-if 4227  df-sn 4318  df-pr 4320  df-op 4324  df-br 4788  df-opab 4848  df-xp 5256  df-rel 5257  df-dm 5260
This theorem is referenced by:  relrn0  5520  coeq0  5787  fnresdisj  6140  fn0  6150  fresaunres2  6217  funopsn  6559  fsnunfv  6600  frxp  7442  domss2  8279  swrd0  13643  setsres  16108  pmtrsn  18146  gsumval3  18515  00lsp  19194  metn0  22385  wlkn0  26751  eulerpath  27421  funresdm1  29754  dfrdg2  32037  mbfresfi  33787  mapfzcons1  37804  diophrw  37846  eldioph2lem1  37847  eldioph2lem2  37848  sge0cl  41110
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