Theorem rel0 5765

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 4366	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5648	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3450   ⊆ wss 3917  ∅c0 4299   × cxp 5639  Rel wrel 5646

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 2008  ax-8 2111  ax-9 2119  ax-ext 2702

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2709  df-cleq 2722  df-clel 2804  df-dif 3920  df-ss 3934  df-nul 4300  df-rel 5648

This theorem is referenced by:  relsnb  5768  reldm0  5894  cnveq0  6173  co02  6236  co01  6237  tpos0  8238  0we1  8473  0er  8712  canthwe  10611  relexpreld  15013  disjALTV0  38753  dibvalrel  41164  dicvalrelN  41186  dihvalrel  41280  reldmprcof1  49374  reldmprcof2  49375  reldmlan2  49610  reldmran2  49611  rellan  49616  relran  49617