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Theorem ltrelpr 6488
Description: Positive real 'less than' is a relation on positive reals. (Contributed by NM, 14-Feb-1996.)
Assertion
Ref Expression
ltrelpr <P ⊆ (P × P)

Proof of Theorem ltrelpr
Dummy variables x 𝑞 y are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-iltp 6453 . 2 <P = {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))}
2 opabssxp 4357 . 2 {⟨x, y⟩ ∣ ((x P y P) 𝑞 Q (𝑞 (2ndx) 𝑞 (1sty)))} ⊆ (P × P)
31, 2eqsstri 2969 1 <P ⊆ (P × P)
Colors of variables: wff set class
Syntax hints:   wa 97   wcel 1390  wrex 2301  wss 2911  {copab 3808   × cxp 4286  cfv 4845  1st c1st 5707  2nd c2nd 5708  Qcnq 6264  Pcnp 6275  <P cltp 6279
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-io 629  ax-5 1333  ax-7 1334  ax-gen 1335  ax-ie1 1379  ax-ie2 1380  ax-8 1392  ax-10 1393  ax-11 1394  ax-i12 1395  ax-bndl 1396  ax-4 1397  ax-17 1416  ax-i9 1420  ax-ial 1424  ax-i5r 1425  ax-ext 2019
This theorem depends on definitions:  df-bi 110  df-nf 1347  df-sb 1643  df-clab 2024  df-cleq 2030  df-clel 2033  df-nfc 2164  df-in 2918  df-ss 2925  df-opab 3810  df-xp 4294  df-iltp 6453
This theorem is referenced by:  ltprordil  6565  ltexprlemm  6574  ltexprlemopl  6575  ltexprlemlol  6576  ltexprlemopu  6577  ltexprlemupu  6578  ltexprlemdisj  6580  ltexprlemloc  6581  ltexprlemfl  6583  ltexprlemrl  6584  ltexprlemfu  6585  ltexprlemru  6586  ltexpri  6587  ltaprlem  6591  gt0srpr  6676  lttrsr  6690  ltposr  6691  archsr  6708
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