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Theorem dedekindeulemlub 12756
Description: Lemma for dedekindeu 12759. The set L has a least upper bound. (Contributed by Jim Kingdon, 31-Jan-2024.)
Hypotheses
Ref Expression
dedekindeu.lss  |-  ( ph  ->  L  C_  RR )
dedekindeu.uss  |-  ( ph  ->  U  C_  RR )
dedekindeu.lm  |-  ( ph  ->  E. q  e.  RR  q  e.  L )
dedekindeu.um  |-  ( ph  ->  E. r  e.  RR  r  e.  U )
dedekindeu.lr  |-  ( ph  ->  A. q  e.  RR  ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
dedekindeu.ur  |-  ( ph  ->  A. r  e.  RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
dedekindeu.disj  |-  ( ph  ->  ( L  i^i  U
)  =  (/) )
dedekindeu.loc  |-  ( ph  ->  A. q  e.  RR  A. r  e.  RR  (
q  <  r  ->  ( q  e.  L  \/  r  e.  U )
) )
Assertion
Ref Expression
dedekindeulemlub  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  L  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  L  y  <  z ) ) )
Distinct variable groups:    L, q, r, x, y, z    U, q, r, y, z    ph, q,
r, x, y, z
Allowed substitution hint:    U( x)

Proof of Theorem dedekindeulemlub
StepHypRef Expression
1 dedekindeu.lss . 2  |-  ( ph  ->  L  C_  RR )
2 dedekindeu.lm . . 3  |-  ( ph  ->  E. q  e.  RR  q  e.  L )
3 eleq1w 2198 . . . . 5  |-  ( q  =  x  ->  (
q  e.  L  <->  x  e.  L ) )
43cbvrexv 2653 . . . 4  |-  ( E. q  e.  RR  q  e.  L  <->  E. x  e.  RR  x  e.  L )
5 rexex 2477 . . . 4  |-  ( E. x  e.  RR  x  e.  L  ->  E. x  x  e.  L )
64, 5sylbi 120 . . 3  |-  ( E. q  e.  RR  q  e.  L  ->  E. x  x  e.  L )
72, 6syl 14 . 2  |-  ( ph  ->  E. x  x  e.  L )
8 dedekindeu.uss . . 3  |-  ( ph  ->  U  C_  RR )
9 dedekindeu.um . . 3  |-  ( ph  ->  E. r  e.  RR  r  e.  U )
10 dedekindeu.lr . . 3  |-  ( ph  ->  A. q  e.  RR  ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
11 dedekindeu.ur . . 3  |-  ( ph  ->  A. r  e.  RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
12 dedekindeu.disj . . 3  |-  ( ph  ->  ( L  i^i  U
)  =  (/) )
13 dedekindeu.loc . . 3  |-  ( ph  ->  A. q  e.  RR  A. r  e.  RR  (
q  <  r  ->  ( q  e.  L  \/  r  e.  U )
) )
141, 8, 2, 9, 10, 11, 12, 13dedekindeulemub 12754 . 2  |-  ( ph  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
151, 8, 2, 9, 10, 11, 12, 13dedekindeulemloc 12755 . 2  |-  ( ph  ->  A. x  e.  RR  A. y  e.  RR  (
x  <  y  ->  ( E. z  e.  L  x  <  z  \/  A. z  e.  L  z  <  y ) ) )
16 axsuploc 7830 . 2  |-  ( ( ( L  C_  RR  /\ 
E. x  x  e.  L )  /\  ( E. x  e.  RR  A. y  e.  L  y  <  x  /\  A. x  e.  RR  A. y  e.  RR  ( x  < 
y  ->  ( E. z  e.  L  x  <  z  \/  A. z  e.  L  z  <  y ) ) ) )  ->  E. x  e.  RR  ( A. y  e.  L  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  L  y  <  z ) ) )
171, 7, 14, 15, 16syl22anc 1217 1  |-  ( ph  ->  E. x  e.  RR  ( A. y  e.  L  -.  x  <  y  /\  A. y  e.  RR  (
y  <  x  ->  E. z  e.  L  y  <  z ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331   E.wex 1468    e. wcel 1480   A.wral 2414   E.wrex 2415    i^i cin 3065    C_ wss 3066   (/)c0 3358   class class class wbr 3924   RRcr 7612    < clt 7793
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-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2119  ax-sep 4041  ax-pow 4093  ax-pr 4126  ax-un 4350  ax-setind 4447  ax-cnex 7704  ax-resscn 7705  ax-pre-ltwlin 7726  ax-pre-suploc 7734
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2000  df-mo 2001  df-clab 2124  df-cleq 2130  df-clel 2133  df-nfc 2268  df-ne 2307  df-nel 2402  df-ral 2419  df-rex 2420  df-rab 2423  df-v 2683  df-dif 3068  df-un 3070  df-in 3072  df-ss 3079  df-nul 3359  df-pw 3507  df-sn 3528  df-pr 3529  df-op 3531  df-uni 3732  df-br 3925  df-opab 3985  df-xp 4540  df-cnv 4542  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799
This theorem is referenced by:  dedekindeulemlu  12757
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