Theorem lerelxr 8135

Description: 'Less than or equal' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
lerelxr	|- <_ C_ ( RR* X. RR* )

Proof of Theorem lerelxr

Step	Hyp	Ref	Expression
1		df-le 8113	. 2 |- <_ = ( ( RR* X. RR* ) \ `' < )
2		difss 3299	. 2 |- ( ( RR* X. RR* ) \ `' < ) C_ ( RR* X. RR* )
3	1, 2	eqsstri 3225	1 |- <_ C_ ( RR* X. RR* )

Syntax hints:   \ cdif 3163   C_ wss 3166   X. cxp 4673   `'ccnv 4674   RR*cxr 8106   < clt 8107   <_ cle 8108

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-in1 615  ax-in2 616  ax-io 711  ax-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-10 1528  ax-11 1529  ax-i12 1530  ax-bndl 1532  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-nfc 2337  df-v 2774  df-dif 3168  df-in 3172  df-ss 3179  df-le 8113

This theorem is referenced by:  lerel  8136  cnfldstr  14320  cnfldle  14329  znval  14398  znle  14399  znbaslemnn  14401