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Theorem ltrelre 6730
Description: 'Less than' is a relation on real numbers. (Contributed by NM, 22-Feb-1996.)
Assertion
Ref Expression
ltrelre < ⊆ (ℝ × ℝ)

Proof of Theorem ltrelre
Dummy variables x y z w are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 df-lt 6724 . 2 < = {⟨x, y⟩ ∣ ((x y ℝ) zw((x = ⟨z, 0R y = ⟨w, 0R⟩) z <R w))}
2 opabssxp 4357 . 2 {⟨x, y⟩ ∣ ((x y ℝ) zw((x = ⟨z, 0R y = ⟨w, 0R⟩) z <R w))} ⊆ (ℝ × ℝ)
31, 2eqsstri 2969 1 < ⊆ (ℝ × ℝ)
Colors of variables: wff set class
Syntax hints:   wa 97   = wceq 1242  wex 1378   wcel 1390  wss 2911  cop 3370   class class class wbr 3755  {copab 3808   × cxp 4286  0Rc0r 6282   <R cltr 6287  cr 6710   < cltrr 6715
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-lt 6724
This theorem is referenced by:  ltresr  6736
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