Theorem rel0 5785

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 4353	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5665	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 234	1 ⊢ Rel ∅

This proof depends on syntax axioms:  Vcvv 3454   ⊆ wss 3900  ∅c0 4281   × cxp 5656  Rel wrel 5663

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 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-tru 1573  df-fal 1583  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-ss 3901  df-dif 4050  df-nul 4282  df-rel 5665

This theorem is used by:  relsnb  5789  reldm0  5922  cnveq0  6209  cnveq0b  6210  co02  6283  co01  6284  tpos0  8311  0we1  8552  0er  8794  canthwe  10775  relexpreld  15236  disjALTV0  40052  dibvalrel  42486  dicvalrelN  42508  dihvalrel  42602  reldmprcof1  50847  reldmprcof2  50848  reldmlan2  51131  reldmran2  51132  rellan  51137  relran  51138