Theorem rel0 5796

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 4394	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5681	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 230	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3475   ⊆ wss 3946  ∅c0 4320   × cxp 5672  Rel wrel 5679

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1545  df-fal 1555  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-dif 3949  df-in 3953  df-ss 3963  df-nul 4321  df-rel 5681

This theorem is referenced by:  relsnb  5799  reldm0  5924  cnveq0  6192  co02  6255  co01  6256  tpos0  8235  0we1  8500  0er  8735  canthwe  10641  relexpreld  14982  disjALTV0  37561  dibvalrel  39971  dicvalrelN  39993  dihvalrel  40087