Theorem rel0 5800

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 4397	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5684	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 230	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3475   ⊆ wss 3949  ∅c0 4323   × cxp 5675  Rel wrel 5682

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 3952  df-in 3956  df-ss 3966  df-nul 4324  df-rel 5684

This theorem is referenced by:  relsnb  5803  reldm0  5928  cnveq0  6197  co02  6260  co01  6261  tpos0  8241  0we1  8506  0er  8740  canthwe  10646  relexpreld  14987  disjALTV0  37624  dibvalrel  40034  dicvalrelN  40056  dihvalrel  40150