Theorem lerelxr 8134

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 8112	. 2 |- <_ = ( ( RR* X. RR* ) \ `' < )
2		difss 3298	. 2 |- ( ( RR* X. RR* ) \ `' < ) C_ ( RR* X. RR* )
3	1, 2	eqsstri 3224	1 |- <_ C_ ( RR* X. RR* )

Syntax hints:   \ cdif 3162   C_ wss 3165   X. cxp 4672   `'ccnv 4673   RR*cxr 8105   < clt 8106   <_ cle 8107

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 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-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-dif 3167  df-in 3171  df-ss 3178  df-le 8112

This theorem is referenced by:  lerel  8135  cnfldstr  14291  cnfldle  14300  znval  14369  znle  14370  znbaslemnn  14372