Theorem rel0 5756

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 5639	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 231	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3442   ⊆ wss 3903  ∅c0 4287   × cxp 5630  Rel wrel 5637

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-tru 1545  df-fal 1555  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-dif 3906  df-ss 3920  df-nul 4288  df-rel 5639

This theorem is referenced by:  relsnb  5759  reldm0  5885  cnveq0  6163  co02  6227  co01  6228  tpos0  8208  0we1  8443  0er  8684  canthwe  10574  relexpreld  14975  disjALTV0  39105  dibvalrel  41539  dicvalrelN  41561  dihvalrel  41655  reldmprcof1  49740  reldmprcof2  49741  reldmlan2  49976  reldmran2  49977  rellan  49982  relran  49983