Theorem rel0 2499

Description: The empty set is a relation.

Assertion

Ref	Expression
rel0	⊢ Rel ∅

Proof of Theorem rel0

Step	Hyp	Ref	Expression
1		0ss 1725	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 2425	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 165	1 ⊢ Rel ∅

Syntax hints:  Vcvv 1348   ⊆ wss 1487  ∅c0 1707   × cxp 2408  Rel wrel 2415

This theorem is referenced by:  reldm0 2550  intirr 2628  cnv0 2633  co02 2663  co01 2664  fn0 2739

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-dif 1489  df-in 1491  df-ss 1492  df-nul 1708  df-rel 2425

metamath.org