Theorem ltrelre 7774

Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)

Assertion

Ref	Expression
ltrelre	|- <RR C_ ( RR X. RR )

Proof of Theorem ltrelre

Dummy variables x y z w are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		df-lt 7766	. 2 |- <RR = { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) }
2		opabssxp 4678	. 2 |- { <. x , y >. | ( ( x e. RR /\ y e. RR ) /\ E. z E. w ( ( x = <. z , 0R >. /\ y = <. w , 0R >. ) /\ z <R w ) ) } C_ ( RR X. RR )
3	1, 2	eqsstri 3174	1 |- <RR C_ ( RR X. RR )

Syntax hints:   /\ wa 103   = wceq 1343   E.wex 1480   e. wcel 2136   C_ wss 3116   <.cop 3579   class class class wbr 3982   {copab 4042   X. cxp 4602   0Rc0r 7239   <R cltr 7244   RRcr 7752   <RR cltrr 7757

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-in 3122  df-ss 3129  df-opab 4044  df-xp 4610  df-lt 7766

This theorem is referenced by:  ltresr  7780