Theorem rel0 5788

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 4360	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5671	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 234	1 ⊢ Rel ∅

This proof depends on syntax axioms:  Vcvv 3458   ⊆ wss 3908  ∅c0 4289   × cxp 5662  Rel wrel 5669

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1943  ax-6 2000  ax-7 2041  ax-8 2148  ax-9 2156  ax-ext 2738

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2100  df-clab 2745  df-cleq 2758  df-clel 2841  df-dif 3911  df-ss 3925  df-nul 4290  df-rel 5671

This theorem is used by:  relsnb  5792  reldm0  5921  cnveq0  6199  co02  6265  co01  6266  tpos0  8254  0we1  8493  0er  8735  canthwe  10646  relexpreld  15088  disjALTV0  39535  dibvalrel  41969  dicvalrelN  41991  dihvalrel  42085  reldmprcof1  50191  reldmprcof2  50192  reldmlan2  50427  reldmran2  50428  rellan  50433  relran  50434