Theorem lerelxr 8089

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 8067	. 2 |- <_ = ( ( RR* X. RR* ) \ `' < )
2		difss 3289	. 2 |- ( ( RR* X. RR* ) \ `' < ) C_ ( RR* X. RR* )
3	1, 2	eqsstri 3215	1 |- <_ C_ ( RR* X. RR* )

Syntax hints:   \ cdif 3154   C_ wss 3157   X. cxp 4661   `'ccnv 4662   RR*cxr 8060   < clt 8061   <_ cle 8062

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-dif 3159  df-in 3163  df-ss 3170  df-le 8067

This theorem is referenced by:  lerel  8090  cnfldstr  14114  cnfldle  14123  znval  14192  znle  14193  znbaslemnn  14195