Theorem ltrelpr 7313

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 7278	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4613	. 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 3129	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 103   e. wcel 1480   E.wrex 2417   C_ wss 3071   {copab 3988   X. cxp 4537   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7088   P.cnp 7099   <P cltp 7103

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-in 3077  df-ss 3084  df-opab 3990  df-xp 4545  df-iltp 7278

This theorem is referenced by:  ltprordil  7397  ltexprlemm  7408  ltexprlemopl  7409  ltexprlemlol  7410  ltexprlemopu  7411  ltexprlemupu  7412  ltexprlemdisj  7414  ltexprlemloc  7415  ltexprlemfl  7417  ltexprlemrl  7418  ltexprlemfu  7419  ltexprlemru  7420  ltexpri  7421  lteupri  7425  ltaprlem  7426  prplnqu  7428  caucvgprprlemk  7491  caucvgprprlemnkltj  7497  caucvgprprlemnkeqj  7498  caucvgprprlemnjltk  7499  caucvgprprlemnbj  7501  caucvgprprlemml  7502  caucvgprprlemlol  7506  caucvgprprlemupu  7508  suplocexprlemss  7523  suplocexprlemlub  7532  gt0srpr  7556  lttrsr  7570  ltposr  7571  archsr  7590