Theorem rel0 5799

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 4396	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5683	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 230	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3474   ⊆ wss 3948  ∅c0 4322   × cxp 5674  Rel wrel 5681

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-dif 3951  df-in 3955  df-ss 3965  df-nul 4323  df-rel 5683

This theorem is referenced by:  relsnb  5802  reldm0  5927  cnveq0  6196  co02  6259  co01  6260  tpos0  8240  0we1  8505  0er  8739  canthwe  10645  relexpreld  14986  disjALTV0  37619  dibvalrel  40029  dicvalrelN  40051  dihvalrel  40145