Theorem ltrelpr 7589

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

Assertion

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

Proof of Theorem ltrelpr

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

Step	Hyp	Ref	Expression
1		df-iltp 7554	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4738	. 2 |- { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) } C_ ( P. X. P. )
3	1, 2	eqsstri 3216	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2167   E.wrex 2476   C_ wss 3157   {copab 4094   X. cxp 4662   ` cfv 5259   1stc1st 6205   2ndc2nd 6206   Q.cnq 7364   P.cnp 7375   <P cltp 7379

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 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-in 3163  df-ss 3170  df-opab 4096  df-xp 4670  df-iltp 7554

This theorem is referenced by:  ltprordil  7673  ltexprlemm  7684  ltexprlemopl  7685  ltexprlemlol  7686  ltexprlemopu  7687  ltexprlemupu  7688  ltexprlemdisj  7690  ltexprlemloc  7691  ltexprlemfl  7693  ltexprlemrl  7694  ltexprlemfu  7695  ltexprlemru  7696  ltexpri  7697  lteupri  7701  ltaprlem  7702  prplnqu  7704  caucvgprprlemk  7767  caucvgprprlemnkltj  7773  caucvgprprlemnkeqj  7774  caucvgprprlemnjltk  7775  caucvgprprlemnbj  7777  caucvgprprlemml  7778  caucvgprprlemlol  7782  caucvgprprlemupu  7784  suplocexprlemss  7799  suplocexprlemlub  7808  gt0srpr  7832  lttrsr  7846  ltposr  7847  archsr  7866