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Theorem dedekindeulemub 12765
Description: Lemma for dedekindeu 12770. The lower cut has an 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
dedekindeulemub  |-  ( ph  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
Distinct variable groups:    L, q, r, y    x, L, r, y    U, q, r, y    ph, q, r, y
Allowed substitution hints:    ph( x)    U( x)

Proof of Theorem dedekindeulemub
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dedekindeu.um . . 3  |-  ( ph  ->  E. r  e.  RR  r  e.  U )
2 eleq1w 2200 . . . 4  |-  ( r  =  a  ->  (
r  e.  U  <->  a  e.  U ) )
32cbvrexv 2655 . . 3  |-  ( E. r  e.  RR  r  e.  U  <->  E. a  e.  RR  a  e.  U )
41, 3sylib 121 . 2  |-  ( ph  ->  E. a  e.  RR  a  e.  U )
5 simprl 520 . . 3  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  -> 
a  e.  RR )
6 dedekindeu.lss . . . . 5  |-  ( ph  ->  L  C_  RR )
76adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  L  C_  RR )
8 dedekindeu.uss . . . . 5  |-  ( ph  ->  U  C_  RR )
98adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  U  C_  RR )
10 dedekindeu.lm . . . . 5  |-  ( ph  ->  E. q  e.  RR  q  e.  L )
1110adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  E. q  e.  RR  q  e.  L )
121adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  E. r  e.  RR  r  e.  U )
13 dedekindeu.lr . . . . 5  |-  ( ph  ->  A. q  e.  RR  ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
1413adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  A. q  e.  RR  ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
15 dedekindeu.ur . . . . 5  |-  ( ph  ->  A. r  e.  RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
1615adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  A. r  e.  RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
17 dedekindeu.disj . . . . 5  |-  ( ph  ->  ( L  i^i  U
)  =  (/) )
1817adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  -> 
( L  i^i  U
)  =  (/) )
19 dedekindeu.loc . . . . 5  |-  ( ph  ->  A. q  e.  RR  A. r  e.  RR  (
q  <  r  ->  ( q  e.  L  \/  r  e.  U )
) )
2019adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  A. q  e.  RR  A. r  e.  RR  (
q  <  r  ->  ( q  e.  L  \/  r  e.  U )
) )
21 simprr 521 . . . 4  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  -> 
a  e.  U )
227, 9, 11, 12, 14, 16, 18, 20, 21dedekindeulemuub 12764 . . 3  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  A. y  e.  L  y  <  a )
23 brralrspcev 3986 . . 3  |-  ( ( a  e.  RR  /\  A. y  e.  L  y  <  a )  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
245, 22, 23syl2anc 408 . 2  |-  ( (
ph  /\  ( a  e.  RR  /\  a  e.  U ) )  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
254, 24rexlimddv 2554 1  |-  ( ph  ->  E. x  e.  RR  A. y  e.  L  y  <  x )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   A.wral 2416   E.wrex 2417    i^i cin 3070    C_ wss 3071   (/)c0 3363   class class class wbr 3929   RRcr 7619    < clt 7800
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 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltwlin 7733
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 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-nul 3364  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-cnv 4547  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806
This theorem is referenced by:  dedekindeulemlub  12767
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