Theorem rel0 5812

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 4406	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5696	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3478   ⊆ wss 3963  ∅c0 4339   × cxp 5687  Rel wrel 5694

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1908  ax-6 1965  ax-7 2005  ax-8 2108  ax-9 2116  ax-ext 2706

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1540  df-fal 1550  df-ex 1777  df-sb 2063  df-clab 2713  df-cleq 2727  df-clel 2814  df-dif 3966  df-ss 3980  df-nul 4340  df-rel 5696

This theorem is referenced by:  relsnb  5815  reldm0  5941  cnveq0  6219  co02  6282  co01  6283  tpos0  8280  0we1  8543  0er  8782  canthwe  10689  relexpreld  15076  disjALTV0  38736  dibvalrel  41146  dicvalrelN  41168  dihvalrel  41262