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Axiom ax-arch 7408
Description: Archimedean axiom. Definition 3.1(2) of [Geuvers], p. 9. Axiom for real and complex numbers, justified by theorem axarch 7370.

This axiom should not be used directly; instead use arch 8603 (which is the same, but stated in terms of 
NN and  <). (Contributed by Jim Kingdon, 2-May-2020.) (New usage is discouraged.)

Assertion
Ref Expression
ax-arch  |-  ( A  e.  RR  ->  E. n  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } A  <RR  n )
Distinct variable group:    A, n, x, y

Detailed syntax breakdown of Axiom ax-arch
StepHypRef Expression
1 cA . . 3  class  A
2 cr 7293 . . 3  class  RR
31, 2wcel 1436 . 2  wff  A  e.  RR
4 vn . . . . 5  setvar  n
54cv 1286 . . . 4  class  n
6 cltrr 7298 . . . 4  class  <RR
71, 5, 6wbr 3820 . . 3  wff  A  <RR  n
8 c1 7295 . . . . . . 7  class  1
9 vx . . . . . . . 8  setvar  x
109cv 1286 . . . . . . 7  class  x
118, 10wcel 1436 . . . . . 6  wff  1  e.  x
12 vy . . . . . . . . . 10  setvar  y
1312cv 1286 . . . . . . . . 9  class  y
14 caddc 7297 . . . . . . . . 9  class  +
1513, 8, 14co 5613 . . . . . . . 8  class  ( y  +  1 )
1615, 10wcel 1436 . . . . . . 7  wff  ( y  +  1 )  e.  x
1716, 12, 10wral 2355 . . . . . 6  wff  A. y  e.  x  ( y  +  1 )  e.  x
1811, 17wa 102 . . . . 5  wff  ( 1  e.  x  /\  A. y  e.  x  (
y  +  1 )  e.  x )
1918, 9cab 2071 . . . 4  class  { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
2019cint 3671 . . 3  class  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) }
217, 4, 20wrex 2356 . 2  wff  E. n  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } A  <RR  n
223, 21wi 4 1  wff  ( A  e.  RR  ->  E. n  e.  |^| { x  |  ( 1  e.  x  /\  A. y  e.  x  ( y  +  1 )  e.  x ) } A  <RR  n )
Colors of variables: wff set class
This axiom is referenced by:  arch  8603
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