Theorem ltrelpr 7820

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 7785	. 2 |- <P = { <. x , y >. | ( ( x e. P. /\ y e. P. ) /\ E. q e. Q. ( q e. ( 2nd ` x ) /\ q e. ( 1st ` y ) ) ) }
2		opabssxp 4824	. 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 3270	1 |- <P C_ ( P. X. P. )

Syntax hints:   /\ wa 104   e. wcel 2203   E.wrex 2521   C_ wss 3211   {copab 4170   X. cxp 4747   ` cfv 5352   1stc1st 6332   2ndc2nd 6333   Q.cnq 7595   P.cnp 7606   <P cltp 7610

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 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-ext 2214

This theorem depends on definitions:  df-bi 117  df-nf 1510  df-sb 1812  df-clab 2219  df-cleq 2225  df-clel 2228  df-nfc 2373  df-in 3217  df-ss 3224  df-opab 4172  df-xp 4755  df-iltp 7785

This theorem is referenced by:  ltprordil  7904  ltexprlemm  7915  ltexprlemopl  7916  ltexprlemlol  7917  ltexprlemopu  7918  ltexprlemupu  7919  ltexprlemdisj  7921  ltexprlemloc  7922  ltexprlemfl  7924  ltexprlemrl  7925  ltexprlemfu  7926  ltexprlemru  7927  ltexpri  7928  lteupri  7932  ltaprlem  7933  prplnqu  7935  caucvgprprlemk  7998  caucvgprprlemnkltj  8004  caucvgprprlemnkeqj  8005  caucvgprprlemnjltk  8006  caucvgprprlemnbj  8008  caucvgprprlemml  8009  caucvgprprlemlol  8013  caucvgprprlemupu  8015  suplocexprlemss  8030  suplocexprlemlub  8039  gt0srpr  8063  lttrsr  8077  ltposr  8078  archsr  8097