Theorem ltrelre 7807

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 7799	. 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 4694	. 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 3185	1 |- <RR C_ ( RR X. RR )

Syntax hints:   /\ wa 104   = wceq 1353   E.wex 1490   e. wcel 2146   C_ wss 3127   <.cop 3592   class class class wbr 3998   {copab 4058   X. cxp 4618   0Rc0r 7272   <R cltr 7277   RRcr 7785   <RR cltrr 7790

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 709  ax-5 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-ext 2157

This theorem depends on definitions:  df-bi 117  df-nf 1459  df-sb 1761  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-in 3133  df-ss 3140  df-opab 4060  df-xp 4626  df-lt 7799

This theorem is referenced by:  ltresr  7813