Theorem rel0 5809

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 4400	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5692	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3480   ⊆ wss 3951  ∅c0 4333   × cxp 5683  Rel wrel 5690

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 2708

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 2715  df-cleq 2729  df-clel 2816  df-dif 3954  df-ss 3968  df-nul 4334  df-rel 5692

This theorem is referenced by:  relsnb  5812  reldm0  5938  cnveq0  6217  co02  6280  co01  6281  tpos0  8281  0we1  8544  0er  8783  canthwe  10691  relexpreld  15079  disjALTV0  38755  dibvalrel  41165  dicvalrelN  41187  dihvalrel  41281