ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  ltrelpr GIF version

Theorem ltrelpr 7336
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 7301 . 2 <P = {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))}
2 opabssxp 4620 . 2 {⟨𝑥, 𝑦⟩ ∣ ((𝑥P𝑦P) ∧ ∃𝑞Q (𝑞 ∈ (2nd𝑥) ∧ 𝑞 ∈ (1st𝑦)))} ⊆ (P × P)
31, 2eqsstri 3133 1 <P ⊆ (P × P)
Colors of variables: wff set class
Syntax hints:  wa 103  wcel 1481  wrex 2418  wss 3075  {copab 3995   × cxp 4544  cfv 5130  1st c1st 6043  2nd c2nd 6044  Qcnq 7111  Pcnp 7122  <P cltp 7126
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 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122
This theorem depends on definitions:  df-bi 116  df-nf 1438  df-sb 1737  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-in 3081  df-ss 3088  df-opab 3997  df-xp 4552  df-iltp 7301
This theorem is referenced by:  ltprordil  7420  ltexprlemm  7431  ltexprlemopl  7432  ltexprlemlol  7433  ltexprlemopu  7434  ltexprlemupu  7435  ltexprlemdisj  7437  ltexprlemloc  7438  ltexprlemfl  7440  ltexprlemrl  7441  ltexprlemfu  7442  ltexprlemru  7443  ltexpri  7444  lteupri  7448  ltaprlem  7449  prplnqu  7451  caucvgprprlemk  7514  caucvgprprlemnkltj  7520  caucvgprprlemnkeqj  7521  caucvgprprlemnjltk  7522  caucvgprprlemnbj  7524  caucvgprprlemml  7525  caucvgprprlemlol  7529  caucvgprprlemupu  7531  suplocexprlemss  7546  suplocexprlemlub  7555  gt0srpr  7579  lttrsr  7593  ltposr  7594  archsr  7613
  Copyright terms: Public domain W3C validator