Theorem ltrelpr 7862

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 7827	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4844	. 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 3280	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2209   E.wrex 2529   C_ wss 3220   {copab 4186   X. cxp 4767   ` cfv 5372   1stc1st 6362   2ndc2nd 6363   Q.cnq 7637   P.cnp 7648   <P cltp 7652

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 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This theorem depends on definitions:  df-bi 117  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-in 3226  df-ss 3233  df-opab 4188  df-xp 4775  df-iltp 7827

This theorem is referenced by:  ltprordil  7946  ltexprlemm  7957  ltexprlemopl  7958  ltexprlemlol  7959  ltexprlemopu  7960  ltexprlemupu  7961  ltexprlemdisj  7963  ltexprlemloc  7964  ltexprlemfl  7966  ltexprlemrl  7967  ltexprlemfu  7968  ltexprlemru  7969  ltexpri  7970  lteupri  7974  ltaprlem  7975  prplnqu  7977  caucvgprprlemk  8040  caucvgprprlemnkltj  8046  caucvgprprlemnkeqj  8047  caucvgprprlemnjltk  8048  caucvgprprlemnbj  8050  caucvgprprlemml  8051  caucvgprprlemlol  8055  caucvgprprlemupu  8057  suplocexprlemss  8072  suplocexprlemlub  8081  gt0srpr  8105  lttrsr  8119  ltposr  8120  archsr  8139