Theorem rel0 5669

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 4311	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5558	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 234	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3408   ⊆ wss 3866  ∅c0 4237   × cxp 5549  Rel wrel 5556

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1803  ax-4 1817  ax-5 1918  ax-6 1976  ax-7 2016  ax-8 2112  ax-9 2120  ax-ext 2708

This theorem depends on definitions:  df-bi 210  df-an 400  df-tru 1546  df-fal 1556  df-ex 1788  df-sb 2071  df-clab 2715  df-cleq 2729  df-clel 2816  df-v 3410  df-dif 3869  df-in 3873  df-ss 3883  df-nul 4238  df-rel 5558

This theorem is referenced by:  relsnb  5672  reldm0  5797  cnveq0  6060  co02  6124  co01  6125  tpos0  7998  0we1  8233  0er  8428  canthwe  10265  relexpreld  14603  disjALTV0  36599  dibvalrel  38914  dicvalrelN  38936  dihvalrel  39030