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Theorem dedekindeulemlub 13991
Description: Lemma for dedekindeu 13994. 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 2238 . . . . 5  |-  ( q  =  x  ->  (
q  e.  L  <->  x  e.  L ) )
43cbvrexv 2704 . . . 4  |-  ( E. q  e.  RR  q  e.  L  <->  E. x  e.  RR  x  e.  L )
5 rexex 2523 . . . 4  |-  ( E. x  e.  RR  x  e.  L  ->  E. x  x  e.  L )
64, 5sylbi 121 . . 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 13989 . 2  |-  ( ph  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
151, 8, 2, 9, 10, 11, 12, 13dedekindeulemloc 13990 . 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 8028 . 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 1239 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 104    <-> wb 105    \/ wo 708    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456    i^i cin 3128    C_ wss 3129   (/)c0 3422   class class class wbr 4003   RRcr 7809    < clt 7990
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4121  ax-pow 4174  ax-pr 4209  ax-un 4433  ax-setind 4536  ax-cnex 7901  ax-resscn 7902  ax-pre-ltwlin 7923  ax-pre-suploc 7931
This theorem depends on definitions:  df-bi 117  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2739  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3577  df-sn 3598  df-pr 3599  df-op 3601  df-uni 3810  df-br 4004  df-opab 4065  df-xp 4632  df-cnv 4634  df-pnf 7992  df-mnf 7993  df-xr 7994  df-ltxr 7995  df-le 7996
This theorem is referenced by:  dedekindeulemlu  13992
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