Theorem rel0 5778

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 4375	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5661	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3459   ⊆ wss 3926  ∅c0 4308   × cxp 5652  Rel wrel 5659

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2065  df-clab 2714  df-cleq 2727  df-clel 2809  df-dif 3929  df-ss 3943  df-nul 4309  df-rel 5661

This theorem is referenced by:  relsnb  5781  reldm0  5907  cnveq0  6186  co02  6249  co01  6250  tpos0  8255  0we1  8518  0er  8757  canthwe  10665  relexpreld  15059  disjALTV0  38772  dibvalrel  41182  dicvalrelN  41204  dihvalrel  41298  reldmprcof1  49291  reldmprcof2  49292  reldmlan2  49492  reldmran2  49493  rellan  49498  relran  49499