Theorem rel0 5166

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 3924	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5045	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 220	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3173   ⊆ wss 3540  ∅c0 3874   × cxp 5036  Rel wrel 5043

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-dif 3543  df-in 3547  df-ss 3554  df-nul 3875  df-rel 5045

This theorem is referenced by:  reldm0  5264  cnv0OLD  5455  cnveq0  5509  co02  5566  co01  5567  tpos0  7269  0we1  7473  0er  7667  0erOLD  7668  canthwe  9352  dibvalrel  35470  dicvalrelN  35492  dihvalrel  35586