Theorem rel0 4850

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 3531	. 2 ⊢ ∅ ⊆ (V × V)
2		df-rel 4730	. 2 ⊢ (Rel ∅ ↔ ∅ ⊆ (V × V))
3	1, 2	mpbir 146	1 ⊢ Rel ∅

Syntax hints:  Vcvv 2800   ⊆ wss 3198  ∅c0 3492   × cxp 4721  Rel wrel 4728

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-ext 2211

This theorem depends on definitions:  df-bi 117  df-tru 1398  df-nf 1507  df-sb 1809  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-v 2802  df-dif 3200  df-in 3204  df-ss 3211  df-nul 3493  df-rel 4730

This theorem is referenced by:  reldm0  4947  cnv0  5138  cnveq0  5191  co02  5248  co01  5249  tpos0  6435  0er  6731