Theorem ltrelre 8148

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 8140	. 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 4824	. 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 3270	1 |- <RR C_ ( RR X. RR )

Syntax hints:   /\ wa 104   = wceq 1398   E.wex 1541   e. wcel 2203   C_ wss 3211   <.cop 3692   class class class wbr 4109   {copab 4170   X. cxp 4747   0Rc0r 7613   <R cltr 7618   RRcr 8126   <RR cltrr 8131

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-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-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224  df-opab 4172  df-xp 4755  df-lt 8140

This theorem is referenced by:  ltresr  8154