Theorem rel0 5698

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 4327	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5587	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 230	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3422   ⊆ wss 3883  ∅c0 4253   × cxp 5578  Rel wrel 5585

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1799  ax-4 1813  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2110  ax-9 2118  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 396  df-tru 1542  df-fal 1552  df-ex 1784  df-sb 2069  df-clab 2716  df-cleq 2730  df-clel 2817  df-v 3424  df-dif 3886  df-in 3890  df-ss 3900  df-nul 4254  df-rel 5587

This theorem is referenced by:  relsnb  5701  reldm0  5826  cnveq0  6089  co02  6153  co01  6154  tpos0  8043  0we1  8298  0er  8493  canthwe  10338  relexpreld  14679  disjALTV0  36789  dibvalrel  39104  dicvalrelN  39126  dihvalrel  39220