Theorem rel0 5771

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 4354	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5654	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 233	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3454   ⊆ wss 3904  ∅c0 4285   × cxp 5645  Rel wrel 5652

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1815  ax-4 1829  ax-5 1930  ax-6 1987  ax-7 2028  ax-8 2144  ax-9 2152  ax-ext 2734

This theorem depends on definitions:  df-bi 209  df-an 400  df-tru 1563  df-fal 1573  df-ex 1800  df-sb 2091  df-clab 2741  df-cleq 2754  df-clel 2837  df-dif 3907  df-ss 3921  df-nul 4286  df-rel 5654

This theorem is referenced by:  relsnb  5775  reldm0  5904  cnveq0  6184  co02  6248  co01  6249  tpos0  8236  0we1  8475  0er  8717  canthwe  10609  relexpreld  15053  disjALTV0  39353  dibvalrel  41787  dicvalrelN  41809  dihvalrel  41903  reldmprcof1  50002  reldmprcof2  50003  reldmlan2  50238  reldmran2  50239  rellan  50244  relran  50245