Theorem rel0 5243

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 3972	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 5121	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 221	1 ⊢ Rel ∅

Syntax hints:  Vcvv 3200   ⊆ wss 3574  ∅c0 3915   × cxp 5112  Rel wrel 5119

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-dif 3577  df-in 3581  df-ss 3588  df-nul 3916  df-rel 5121

This theorem is referenced by:  reldm0  5343  cnv0OLD  5536  cnveq0  5591  co02  5649  co01  5650  tpos0  7382  0we1  7586  0er  7780  0erOLD  7781  canthwe  9473  opabf  34131  dibvalrel  36452  dicvalrelN  36474  dihvalrel  36568