Theorem rel0 5753

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 4359	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5638	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3444   ⊆ wss 3911  ∅c0 4292   × cxp 5629  Rel wrel 5636

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1543  df-fal 1553  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-dif 3914  df-ss 3928  df-nul 4293  df-rel 5638

This theorem is referenced by:  relsnb  5756  reldm0  5881  cnveq0  6158  co02  6221  co01  6222  tpos0  8212  0we1  8447  0er  8686  canthwe  10580  relexpreld  14982  disjALTV0  38719  dibvalrel  41130  dicvalrelN  41152  dihvalrel  41246  reldmprcof1  49343  reldmprcof2  49344  reldmlan2  49579  reldmran2  49580  rellan  49585  relran  49586