Theorem rel0 5667

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 4350	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5557	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 233	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3495   ⊆ wss 3936  ∅c0 4291   × cxp 5548  Rel wrel 5555

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1792  ax-4 1806  ax-5 1907  ax-6 1966  ax-7 2011  ax-8 2112  ax-9 2120  ax-10 2141  ax-11 2156  ax-12 2172  ax-ext 2793

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1536  df-ex 1777  df-nf 1781  df-sb 2066  df-clab 2800  df-cleq 2814  df-clel 2893  df-dif 3939  df-in 3943  df-ss 3952  df-nul 4292  df-rel 5557

This theorem is referenced by:  relsnb  5670  reldm0  5793  cnveq0  6049  co02  6108  co01  6109  tpos0  7916  0we1  8125  0er  8320  canthwe  10067  disjALTV0  35978  dibvalrel  38293  dicvalrelN  38315  dihvalrel  38409