Theorem ltrelpr 7591

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 7556	. 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 7366   P.cnp 7377   <P cltp 7381

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 7556

This theorem is referenced by:  ltprordil  7675  ltexprlemm  7686  ltexprlemopl  7687  ltexprlemlol  7688  ltexprlemopu  7689  ltexprlemupu  7690  ltexprlemdisj  7692  ltexprlemloc  7693  ltexprlemfl  7695  ltexprlemrl  7696  ltexprlemfu  7697  ltexprlemru  7698  ltexpri  7699  lteupri  7703  ltaprlem  7704  prplnqu  7706  caucvgprprlemk  7769  caucvgprprlemnkltj  7775  caucvgprprlemnkeqj  7776  caucvgprprlemnjltk  7777  caucvgprprlemnbj  7779  caucvgprprlemml  7780  caucvgprprlemlol  7784  caucvgprprlemupu  7786  suplocexprlemss  7801  suplocexprlemlub  7810  gt0srpr  7834  lttrsr  7848  ltposr  7849  archsr  7868