Theorem ltrel 8015

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 8014	. 2 |- < C_ ( RR* X. RR* )
2		relxp 4734	. 2 |- Rel ( RR* X. RR* )
3		relss 4712	. 2 |- ( < C_ ( RR* X. RR* ) -> ( Rel ( RR* X. RR* ) -> Rel < ) )
4	1, 2, 3	mp2 16	1 |- Rel <

Syntax hints:   C_ wss 3129   X. cxp 4623   Rel wrel 4630   RR*cxr 7987   < clt 7988

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 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-ext 2159

This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-nf 1461  df-sb 1763  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-v 2739  df-un 3133  df-in 3135  df-ss 3142  df-pr 3599  df-opab 4064  df-xp 4631  df-rel 4632  df-xr 7992  df-ltxr 7993

This theorem is referenced by: (None)