Theorem ltrelre 7795

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 7787	. 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 4685	. 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 3179	1 |- <RR C_ ( RR X. RR )

Syntax hints:   /\ wa 103   = wceq 1348   E.wex 1485   e. wcel 2141   C_ wss 3121   <.cop 3586   class class class wbr 3989   {copab 4049   X. cxp 4609   0Rc0r 7260   <R cltr 7265   RRcr 7773   <RR cltrr 7778

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 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-in 3127  df-ss 3134  df-opab 4051  df-xp 4617  df-lt 7787

This theorem is referenced by:  ltresr  7801