Theorem ltrelpr 9820

Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.) (New usage is discouraged.)

Assertion

Ref	Expression
ltrelpr	|- <P C_ ( P. X. P. )

Proof of Theorem ltrelpr

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

Step	Hyp	Ref	Expression
1		df-ltp 9807	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ x C. y ) }
2		opabssxp 5193	. 2 |- { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ x C. y ) } C_ ( P. X. P. )
3	1, 2	eqsstri 3635	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 384   e. wcel 1990   C_ wss 3574   C. wpss 3575   {copab 4712   X. cxp 5112   P.cnp 9681   <P cltp 9685

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-in 3581  df-ss 3588  df-opab 4713  df-xp 5120  df-ltp 9807

This theorem is referenced by:  ltexpri  9865  ltaprlem  9866  ltapr  9867  suplem1pr  9874  suplem2pr  9875  supexpr  9876  ltsrpr  9898  ltsosr  9915  mappsrpr  9929