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Theorem dedekindicclemub 12774
Description: Lemma for dedekindicc 12780. The lower cut has an upper bound. (Contributed by Jim Kingdon, 15-Feb-2024.)
Hypotheses
Ref Expression
dedekindicc.a  |-  ( ph  ->  A  e.  RR )
dedekindicc.b  |-  ( ph  ->  B  e.  RR )
dedekindicc.lss  |-  ( ph  ->  L  C_  ( A [,] B ) )
dedekindicc.uss  |-  ( ph  ->  U  C_  ( A [,] B ) )
dedekindicc.lm  |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )
dedekindicc.um  |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )
dedekindicc.lr  |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
dedekindicc.ur  |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
dedekindicc.disj  |-  ( ph  ->  ( L  i^i  U
)  =  (/) )
dedekindicc.loc  |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U
) ) )
Assertion
Ref Expression
dedekindicclemub  |-  ( ph  ->  E. x  e.  ( A [,] B ) A. y  e.  L  y  <  x )
Distinct variable groups:    A, q, r, y    x, A, y    B, q, r, y    x, B    L, q, y    x, L    U, q, r, y    ph, q, y
Allowed substitution hints:    ph( x, r)    U( x)    L( r)

Proof of Theorem dedekindicclemub
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dedekindicc.um . . 3  |-  ( ph  ->  E. r  e.  ( A [,] B ) r  e.  U )
2 eleq1w 2200 . . . 4  |-  ( r  =  a  ->  (
r  e.  U  <->  a  e.  U ) )
32cbvrexv 2655 . . 3  |-  ( E. r  e.  ( A [,] B ) r  e.  U  <->  E. a  e.  ( A [,] B
) a  e.  U
)
41, 3sylib 121 . 2  |-  ( ph  ->  E. a  e.  ( A [,] B ) a  e.  U )
5 simprl 520 . . 3  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  -> 
a  e.  ( A [,] B ) )
6 dedekindicc.a . . . . 5  |-  ( ph  ->  A  e.  RR )
76adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  A  e.  RR )
8 dedekindicc.b . . . . 5  |-  ( ph  ->  B  e.  RR )
98adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  B  e.  RR )
10 dedekindicc.lss . . . . 5  |-  ( ph  ->  L  C_  ( A [,] B ) )
1110adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  L  C_  ( A [,] B ) )
12 dedekindicc.uss . . . . 5  |-  ( ph  ->  U  C_  ( A [,] B ) )
1312adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  U  C_  ( A [,] B ) )
14 dedekindicc.lm . . . . 5  |-  ( ph  ->  E. q  e.  ( A [,] B ) q  e.  L )
1514adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  E. q  e.  ( A [,] B ) q  e.  L )
161adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  E. r  e.  ( A [,] B ) r  e.  U )
17 dedekindicc.lr . . . . 5  |-  ( ph  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
1817adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  A. q  e.  ( A [,] B ) ( q  e.  L  <->  E. r  e.  L  q  <  r ) )
19 dedekindicc.ur . . . . 5  |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
2019adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
21 dedekindicc.disj . . . . 5  |-  ( ph  ->  ( L  i^i  U
)  =  (/) )
2221adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  -> 
( L  i^i  U
)  =  (/) )
23 dedekindicc.loc . . . . 5  |-  ( ph  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  ( q  e.  L  \/  r  e.  U
) ) )
2423adantr 274 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  A. q  e.  ( A [,] B ) A. r  e.  ( A [,] B ) ( q  <  r  ->  (
q  e.  L  \/  r  e.  U )
) )
25 simprr 521 . . . 4  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  -> 
a  e.  U )
267, 9, 11, 13, 15, 16, 18, 20, 22, 24, 25dedekindicclemuub 12773 . . 3  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  A. y  e.  L  y  <  a )
27 brralrspcev 3986 . . 3  |-  ( ( a  e.  ( A [,] B )  /\  A. y  e.  L  y  <  a )  ->  E. x  e.  ( A [,] B ) A. y  e.  L  y  <  x )
285, 26, 27syl2anc 408 . 2  |-  ( (
ph  /\  ( a  e.  ( A [,] B
)  /\  a  e.  U ) )  ->  E. x  e.  ( A [,] B ) A. y  e.  L  y  <  x )
294, 28rexlimddv 2554 1  |-  ( ph  ->  E. x  e.  ( A [,] B ) 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  (class class class)co 5774   RRcr 7619    < clt 7800   [,]cicc 9674
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-ltirr 7732  ax-pre-ltwlin 7733  ax-pre-lttrn 7734
This theorem depends on definitions:  df-bi 116  df-3or 963  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-sbc 2910  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-id 4215  df-po 4218  df-iso 4219  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-iota 5088  df-fun 5125  df-fv 5131  df-ov 5777  df-oprab 5778  df-mpo 5779  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805  df-le 7806  df-icc 9678
This theorem is referenced by: (None)
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