Theorem ltrel 8133

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 8132	. 2 |- < C_ ( RR* X. RR* )
2		relxp 4783	. 2 |- Rel ( RR* X. RR* )
3		relss 4761	. 2 |- ( < C_ ( RR* X. RR* ) -> ( Rel ( RR* X. RR* ) -> Rel < ) )
4	1, 2, 3	mp2 16	1 |- Rel <

Syntax hints:   C_ wss 3165   X. cxp 4672   Rel wrel 4679   RR*cxr 8105   < clt 8106

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 710  ax-5 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-in 3171  df-ss 3178  df-pr 3639  df-opab 4105  df-xp 4680  df-rel 4681  df-xr 8110  df-ltxr 8111

This theorem is referenced by: (None)