Theorem rel0 5709

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 4330	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5596	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 230	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3432   ⊆ wss 3887  ∅c0 4256   × cxp 5587  Rel wrel 5594

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2709

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1542  df-fal 1552  df-ex 1783  df-sb 2068  df-clab 2716  df-cleq 2730  df-clel 2816  df-v 3434  df-dif 3890  df-in 3894  df-ss 3904  df-nul 4257  df-rel 5596

This theorem is referenced by:  relsnb  5712  reldm0  5837  cnveq0  6100  co02  6164  co01  6165  tpos0  8072  0we1  8336  0er  8535  canthwe  10407  relexpreld  14751  disjALTV0  36862  dibvalrel  39177  dicvalrelN  39199  dihvalrel  39293