Theorem rel0 5823

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 4423	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5707	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3488   ⊆ wss 3976  ∅c0 4352   × cxp 5698  Rel wrel 5705

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1793  ax-4 1807  ax-5 1909  ax-6 1967  ax-7 2007  ax-8 2110  ax-9 2118  ax-ext 2711

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1778  df-sb 2065  df-clab 2718  df-cleq 2732  df-clel 2819  df-dif 3979  df-ss 3993  df-nul 4353  df-rel 5707

This theorem is referenced by:  relsnb  5826  reldm0  5952  cnveq0  6228  co02  6291  co01  6292  tpos0  8297  0we1  8562  0er  8801  canthwe  10720  relexpreld  15089  disjALTV0  38710  dibvalrel  41120  dicvalrelN  41142  dihvalrel  41236