Theorem rel0 5787

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 4357	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5670	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 234	1 ⊢ Rel ∅

This proof depends on syntax axioms:  Vcvv 3457   ⊆ wss 3906  ∅c0 4286   × cxp 5661  Rel wrel 5668

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 2737

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 2744  df-cleq 2757  df-clel 2840  df-dif 3909  df-ss 3923  df-nul 4287  df-rel 5670

This theorem is used by:  relsnb  5791  reldm0  5920  cnveq0  6198  co02  6264  co01  6265  tpos0  8258  0we1  8497  0er  8739  canthwe  10651  relexpreld  15101  disjALTV0  39561  dibvalrel  41995  dicvalrelN  42017  dihvalrel  42111  reldmprcof1  50216  reldmprcof2  50217  reldmlan2  50452  reldmran2  50453  rellan  50458  relran  50459