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Theorem reldm0 5481
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 5382 . . 3 Rel ∅
2 eqrel 5349 . . 3 ((Rel 𝐴 ∧ Rel ∅) → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
31, 2mpan2 663 . 2 (Rel 𝐴 → (𝐴 = ∅ ↔ ∀𝑥𝑦(⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅)))
4 eq0 4074 . . 3 (dom 𝐴 = ∅ ↔ ∀𝑥 ¬ 𝑥 ∈ dom 𝐴)
5 alnex 1853 . . . . . 6 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
6 vex 3352 . . . . . . 7 𝑥 ∈ V
76eldm2 5460 . . . . . 6 (𝑥 ∈ dom 𝐴 ↔ ∃𝑦𝑥, 𝑦⟩ ∈ 𝐴)
85, 7xchbinxr 324 . . . . 5 (∀𝑦 ¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ¬ 𝑥 ∈ dom 𝐴)
9 noel 4065 . . . . . . 7 ¬ ⟨𝑥, 𝑦⟩ ∈ ∅
109nbn 361 . . . . . 6 (¬ ⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ (⟨𝑥, 𝑦⟩ ∈ 𝐴 ↔ ⟨𝑥, 𝑦⟩ ∈ ∅))
1110albii 1894 . . . . 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 1894 . . 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 1628   = wceq 1630  wex 1851  wcel 2144  c0 4061  cop 4320  dom cdm 5249  Rel wrel 5254
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1869  ax-4 1884  ax-5 1990  ax-6 2056  ax-7 2092  ax-9 2153  ax-10 2173  ax-11 2189  ax-12 2202  ax-13 2407  ax-ext 2750
This theorem depends on definitions:  df-bi 197  df-an 383  df-or 827  df-3an 1072  df-tru 1633  df-ex 1852  df-nf 1857  df-sb 2049  df-clab 2757  df-cleq 2763  df-clel 2766  df-nfc 2901  df-rab 3069  df-v 3351  df-dif 3724  df-un 3726  df-in 3728  df-ss 3735  df-nul 4062  df-if 4224  df-sn 4315  df-pr 4317  df-op 4321  df-br 4785  df-opab 4845  df-xp 5255  df-rel 5256  df-dm 5259
This theorem is referenced by:  relrn0  5521  coeq0  5788  fnresdisj  6141  fn0  6151  fresaunres2  6216  funopsn  6555  fsnunfv  6596  frxp  7437  domss2  8274  swrd0  13642  setsres  16107  pmtrsn  18145  gsumval3  18514  00lsp  19193  metn0  22384  wlkn0  26750  eulerpath  27418  funresdm1  29748  dfrdg2  32031  mbfresfi  33781  mapfzcons1  37799  diophrw  37841  eldioph2lem1  37842  eldioph2lem2  37843  sge0cl  41109
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