MPE Home Metamath Proof Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >  df-lt Unicode version

Definition df-lt 8750
Description: Define 'less than' on the real subset of complex numbers. Proofs should typically use  < instead; see df-ltxr 8872. (Contributed by NM, 22-Feb-1996.) (New usage is discouraged.)
Assertion
Ref Expression
df-lt  |-  <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
Distinct variable group:    x, y, z, w

Detailed syntax breakdown of Definition df-lt
StepHypRef Expression
1 cltrr 8741 . 2  class  <RR
2 vx . . . . . . 7  set  x
32cv 1622 . . . . . 6  class  x
4 cr 8736 . . . . . 6  class  RR
53, 4wcel 1684 . . . . 5  wff  x  e.  RR
6 vy . . . . . . 7  set  y
76cv 1622 . . . . . 6  class  y
87, 4wcel 1684 . . . . 5  wff  y  e.  RR
95, 8wa 358 . . . 4  wff  ( x  e.  RR  /\  y  e.  RR )
10 vz . . . . . . . . . . 11  set  z
1110cv 1622 . . . . . . . . . 10  class  z
12 c0r 8490 . . . . . . . . . 10  class  0R
1311, 12cop 3643 . . . . . . . . 9  class  <. z ,  0R >.
143, 13wceq 1623 . . . . . . . 8  wff  x  = 
<. z ,  0R >.
15 vw . . . . . . . . . . 11  set  w
1615cv 1622 . . . . . . . . . 10  class  w
1716, 12cop 3643 . . . . . . . . 9  class  <. w ,  0R >.
187, 17wceq 1623 . . . . . . . 8  wff  y  = 
<. w ,  0R >.
1914, 18wa 358 . . . . . . 7  wff  ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )
20 cltr 8495 . . . . . . . 8  class  <R
2111, 16, 20wbr 4023 . . . . . . 7  wff  z  <R  w
2219, 21wa 358 . . . . . 6  wff  ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w )
2322, 15wex 1528 . . . . 5  wff  E. w
( ( x  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w )
2423, 10wex 1528 . . . 4  wff  E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w )
259, 24wa 358 . . 3  wff  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w
( ( x  = 
<. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) )
2625, 2, 6copab 4076 . 2  class  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
271, 26wceq 1623 1  wff  <RR  =  { <. x ,  y >.  |  ( ( x  e.  RR  /\  y  e.  RR )  /\  E. z E. w ( ( x  =  <. z ,  0R >.  /\  y  =  <. w ,  0R >. )  /\  z  <R  w ) ) }
Colors of variables: wff set class
This definition is referenced by:  ltrelre  8756  ltresr  8762
  Copyright terms: Public domain W3C validator