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Axiom ax-pre-sup 8815
Description: A non-empty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by theorem axpre-sup 8791. Note: Normally new proofs would use axsup 8898. (New usage is discouraged.) (Contributed by NM, 13-Oct-2005.)
Assertion
Ref Expression
ax-pre-sup  |-  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
Distinct variable group:    x, y, z, A

Detailed syntax breakdown of Axiom ax-pre-sup
StepHypRef Expression
1 cA . . . 4  class  A
2 cr 8736 . . . 4  class  RR
31, 2wss 3152 . . 3  wff  A  C_  RR
4 c0 3455 . . . 4  class  (/)
51, 4wne 2446 . . 3  wff  A  =/=  (/)
6 vy . . . . . . 7  set  y
76cv 1622 . . . . . 6  class  y
8 vx . . . . . . 7  set  x
98cv 1622 . . . . . 6  class  x
10 cltrr 8741 . . . . . 6  class  <RR
117, 9, 10wbr 4023 . . . . 5  wff  y  <RR  x
1211, 6, 1wral 2543 . . . 4  wff  A. y  e.  A  y  <RR  x
1312, 8, 2wrex 2544 . . 3  wff  E. x  e.  RR  A. y  e.  A  y  <RR  x
143, 5, 13w3a 934 . 2  wff  ( A 
C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x )
159, 7, 10wbr 4023 . . . . . 6  wff  x  <RR  y
1615wn 3 . . . . 5  wff  -.  x  <RR  y
1716, 6, 1wral 2543 . . . 4  wff  A. y  e.  A  -.  x  <RR  y
18 vz . . . . . . . . 9  set  z
1918cv 1622 . . . . . . . 8  class  z
207, 19, 10wbr 4023 . . . . . . 7  wff  y  <RR  z
2120, 18, 1wrex 2544 . . . . . 6  wff  E. z  e.  A  y  <RR  z
2211, 21wi 4 . . . . 5  wff  ( y 
<RR  x  ->  E. z  e.  A  y  <RR  z )
2322, 6, 2wral 2543 . . . 4  wff  A. y  e.  RR  ( y  <RR  x  ->  E. z  e.  A  y  <RR  z )
2417, 23wa 358 . . 3  wff  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) )
2524, 8, 2wrex 2544 . 2  wff  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  ( y  <RR  x  ->  E. z  e.  A  y  <RR  z ) )
2614, 25wi 4 1  wff  ( ( A  C_  RR  /\  A  =/=  (/)  /\  E. x  e.  RR  A. y  e.  A  y  <RR  x )  ->  E. x  e.  RR  ( A. y  e.  A  -.  x  <RR  y  /\  A. y  e.  RR  (
y  <RR  x  ->  E. z  e.  A  y  <RR  z ) ) )
Colors of variables: wff set class
This axiom is referenced by:  axsup  8898
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