Theorem rel0 5518

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 4230	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5410	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 223	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3408   ⊆ wss 3822  ∅c0 4172   × cxp 5401  Rel wrel 5408

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1759  ax-4 1773  ax-5 1870  ax-6 1929  ax-7 1966  ax-8 2053  ax-9 2060  ax-10 2080  ax-11 2094  ax-12 2107  ax-ext 2743

This theorem depends on definitions:  df-bi 199  df-an 388  df-or 835  df-tru 1511  df-ex 1744  df-nf 1748  df-sb 2017  df-clab 2752  df-cleq 2764  df-clel 2839  df-dif 3825  df-in 3829  df-ss 3836  df-nul 4173  df-rel 5410

This theorem is referenced by:  relsnb  5521  reldm0  5638  cnveq0  5890  co02  5949  co01  5950  tpos0  7723  0we1  7931  0er  8124  canthwe  9869  disjALTV0  35466  dibvalrel  37781  dicvalrelN  37803  dihvalrel  37897