Theorem ltrelpr 7572

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 7537	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4737	. 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 3215	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2167   E.wrex 2476   C_ wss 3157   {copab 4093   X. cxp 4661   ` cfv 5258   1stc1st 6196   2ndc2nd 6197   Q.cnq 7347   P.cnp 7358   <P cltp 7362

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 4095  df-xp 4669  df-iltp 7537

This theorem is referenced by:  ltprordil  7656  ltexprlemm  7667  ltexprlemopl  7668  ltexprlemlol  7669  ltexprlemopu  7670  ltexprlemupu  7671  ltexprlemdisj  7673  ltexprlemloc  7674  ltexprlemfl  7676  ltexprlemrl  7677  ltexprlemfu  7678  ltexprlemru  7679  ltexpri  7680  lteupri  7684  ltaprlem  7685  prplnqu  7687  caucvgprprlemk  7750  caucvgprprlemnkltj  7756  caucvgprprlemnkeqj  7757  caucvgprprlemnjltk  7758  caucvgprprlemnbj  7760  caucvgprprlemml  7761  caucvgprprlemlol  7765  caucvgprprlemupu  7767  suplocexprlemss  7782  suplocexprlemlub  7791  gt0srpr  7815  lttrsr  7829  ltposr  7830  archsr  7849