Theorem ltrel 8377

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 8376	. 2 |- < C_ ( RR* X. RR* )
2		relxp 4879	. 2 |- Rel ( RR* X. RR* )
3		relss 4857	. 2 |- ( < C_ ( RR* X. RR* ) -> ( Rel ( RR* X. RR* ) -> Rel < ) )
4	1, 2, 3	mp2 16	1 |- Rel <

Syntax hints:   C_ wss 3220   X. cxp 4767   Rel wrel 4774   RR*cxr 8349   < clt 8350

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-v 2823  df-un 3224  df-in 3226  df-ss 3233  df-pr 3712  df-opab 4188  df-xp 4775  df-rel 4776  df-xr 8354  df-ltxr 8355

This theorem is referenced by: (None)