Theorem ltrel 8240

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 8239	. 2 |- < C_ ( RR* X. RR* )
2		relxp 4835	. 2 |- Rel ( RR* X. RR* )
3		relss 4813	. 2 |- ( < C_ ( RR* X. RR* ) -> ( Rel ( RR* X. RR* ) -> Rel < ) )
4	1, 2, 3	mp2 16	1 |- Rel <

Syntax hints:   C_ wss 3200   X. cxp 4723   Rel wrel 4730   RR*cxr 8212   < clt 8213

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-io 716  ax-5 1495  ax-7 1496  ax-gen 1497  ax-ie1 1541  ax-ie2 1542  ax-8 1552  ax-10 1553  ax-11 1554  ax-i12 1555  ax-bndl 1557  ax-4 1558  ax-17 1574  ax-i9 1578  ax-ial 1582  ax-i5r 1583  ax-ext 2213

This theorem depends on definitions:  df-bi 117  df-3an 1006  df-tru 1400  df-nf 1509  df-sb 1811  df-clab 2218  df-cleq 2224  df-clel 2227  df-nfc 2363  df-v 2804  df-un 3204  df-in 3206  df-ss 3213  df-pr 3676  df-opab 4151  df-xp 4731  df-rel 4732  df-xr 8217  df-ltxr 8218

This theorem is referenced by: (None)