Theorem ltrel 7981

Description: 'Less than' is a relation. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
ltrel	|- Rel <

Proof of Theorem ltrel

Step	Hyp	Ref	Expression
1		ltrelxr 7980	. 2 |- < C_ ( RR* X. RR* )
2		relxp 4720	. 2 |- Rel ( RR* X. RR* )
3		relss 4698	. 2 |- ( < C_ ( RR* X. RR* ) -> ( Rel ( RR* X. RR* ) -> Rel < ) )
4	1, 2, 3	mp2 16	1 |- Rel <

Syntax hints:   C_ wss 3121   X. cxp 4609   Rel wrel 4616   RR*cxr 7953   < clt 7954

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-v 2732  df-un 3125  df-in 3127  df-ss 3134  df-pr 3590  df-opab 4051  df-xp 4617  df-rel 4618  df-xr 7958  df-ltxr 7959

This theorem is referenced by: (None)