Theorem lerelxr 8336

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 8314	. 2 |- <_ = ( ( RR* X. RR* ) \ `' < )
2		difss 3345	. 2 |- ( ( RR* X. RR* ) \ `' < ) C_ ( RR* X. RR* )
3	1, 2	eqsstri 3270	1 |- <_ C_ ( RR* X. RR* )

Syntax hints:   \ cdif 3208   C_ wss 3211   X. cxp 4747   `'ccnv 4748   RR*cxr 8307   < clt 8308   <_ cle 8309

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-tru 1401  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-v 2815  df-dif 3213  df-in 3217  df-ss 3224  df-le 8314

This theorem is referenced by:  lerel  8337  cnfldstr  14706  cnfldle  14715  znval  14784  znle  14785  znbaslemnn  14787