Theorem rel0 5755

Description: The empty set is a relation. (Contributed by NM, 26-Apr-1998.)

Assertion

Ref	Expression
rel0	⊢ Rel ∅

Proof of Theorem rel0

Step	Hyp	Ref	Expression
1		0ss 4340	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5638	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3429   ⊆ wss 3889  ∅c0 4273   × cxp 5629  Rel wrel 5636

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 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2708

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2715  df-cleq 2728  df-clel 2811  df-dif 3892  df-ss 3906  df-nul 4274  df-rel 5638

This theorem is referenced by:  relsnb  5758  reldm0  5883  cnveq0  6161  co02  6225  co01  6226  tpos0  8206  0we1  8441  0er  8682  canthwe  10574  relexpreld  15002  disjALTV0  39175  dibvalrel  41609  dicvalrelN  41631  dihvalrel  41725  reldmprcof1  49856  reldmprcof2  49857  reldmlan2  50092  reldmran2  50093  rellan  50098  relran  50099