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Theorem dedekindicclemuub 12762
Description: Lemma for dedekindicc 12769. Any element of the upper cut is an upper bound for the lower cut. (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
) ) )
dedekindicclemuub.u  |-  ( ph  ->  C  e.  U )
Assertion
Ref Expression
dedekindicclemuub  |-  ( ph  ->  A. z  e.  L  z  <  C )
Distinct variable groups:    A, r    B, r    C, q, r, z    L, q, z    U, q, z, r    ph, q,
z
Allowed substitution hints:    ph( r)    A( z,
q)    B( z, q)    L( r)

Proof of Theorem dedekindicclemuub
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dedekindicclemuub.u . . 3  |-  ( ph  ->  C  e.  U )
2 eleq1 2200 . . . . 5  |-  ( r  =  C  ->  (
r  e.  U  <->  C  e.  U ) )
3 breq2 3928 . . . . . 6  |-  ( r  =  C  ->  (
q  <  r  <->  q  <  C ) )
43rexbidv 2436 . . . . 5  |-  ( r  =  C  ->  ( E. q  e.  U  q  <  r  <->  E. q  e.  U  q  <  C ) )
52, 4bibi12d 234 . . . 4  |-  ( r  =  C  ->  (
( r  e.  U  <->  E. q  e.  U  q  <  r )  <->  ( C  e.  U  <->  E. q  e.  U  q  <  C ) ) )
6 dedekindicc.ur . . . 4  |-  ( ph  ->  A. r  e.  ( A [,] B ) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
7 dedekindicc.uss . . . . 5  |-  ( ph  ->  U  C_  ( A [,] B ) )
87, 1sseldd 3093 . . . 4  |-  ( ph  ->  C  e.  ( A [,] B ) )
95, 6, 8rspcdva 2789 . . 3  |-  ( ph  ->  ( C  e.  U  <->  E. q  e.  U  q  <  C ) )
101, 9mpbid 146 . 2  |-  ( ph  ->  E. q  e.  U  q  <  C )
11 dedekindicc.a . . . . . . 7  |-  ( ph  ->  A  e.  RR )
12 dedekindicc.b . . . . . . 7  |-  ( ph  ->  B  e.  RR )
13 iccssre 9731 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1411, 12, 13syl2anc 408 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
1514ad2antrr 479 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  ( A [,] B )  C_  RR )
16 dedekindicc.lss . . . . . . 7  |-  ( ph  ->  L  C_  ( A [,] B ) )
1716ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  L  C_  ( A [,] B ) )
18 simpr 109 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  e.  L )
1917, 18sseldd 3093 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  e.  ( A [,] B ) )
2015, 19sseldd 3093 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  e.  RR )
217ad2antrr 479 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  U  C_  ( A [,] B ) )
22 simplrl 524 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  e.  U )
2321, 22sseldd 3093 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  e.  ( A [,] B ) )
2415, 23sseldd 3093 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  e.  RR )
2514, 8sseldd 3093 . . . . 5  |-  ( ph  ->  C  e.  RR )
2625ad2antrr 479 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  C  e.  RR )
27 breq1 3927 . . . . . . . . . 10  |-  ( a  =  q  ->  (
a  <  z  <->  q  <  z ) )
2827rspcev 2784 . . . . . . . . 9  |-  ( ( q  e.  U  /\  q  <  z )  ->  E. a  e.  U  a  <  z )
2922, 28sylan 281 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  E. a  e.  U  a  <  z )
3027cbvrexv 2653 . . . . . . . 8  |-  ( E. a  e.  U  a  <  z  <->  E. q  e.  U  q  <  z )
3129, 30sylib 121 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  E. q  e.  U  q  <  z )
32 eleq1 2200 . . . . . . . . 9  |-  ( r  =  z  ->  (
r  e.  U  <->  z  e.  U ) )
33 breq2 3928 . . . . . . . . . 10  |-  ( r  =  z  ->  (
q  <  r  <->  q  <  z ) )
3433rexbidv 2436 . . . . . . . . 9  |-  ( r  =  z  ->  ( E. q  e.  U  q  <  r  <->  E. q  e.  U  q  <  z ) )
3532, 34bibi12d 234 . . . . . . . 8  |-  ( r  =  z  ->  (
( r  e.  U  <->  E. q  e.  U  q  <  r )  <->  ( z  e.  U  <->  E. q  e.  U  q  <  z ) ) )
366ad3antrrr 483 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  A. r  e.  ( A [,] B
) ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
3719adantr 274 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  z  e.  ( A [,] B
) )
3835, 36, 37rspcdva 2789 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  (
z  e.  U  <->  E. q  e.  U  q  <  z ) )
3931, 38mpbird 166 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  z  e.  U )
40 simplll 522 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  ph )
4118adantr 274 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  z  e.  L )
42 dedekindicc.disj . . . . . . . . 9  |-  ( ph  ->  ( L  i^i  U
)  =  (/) )
43 disj 3406 . . . . . . . . 9  |-  ( ( L  i^i  U )  =  (/)  <->  A. z  e.  L  -.  z  e.  U
)
4442, 43sylib 121 . . . . . . . 8  |-  ( ph  ->  A. z  e.  L  -.  z  e.  U
)
4544r19.21bi 2518 . . . . . . 7  |-  ( (
ph  /\  z  e.  L )  ->  -.  z  e.  U )
4640, 41, 45syl2anc 408 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  -.  z  e.  U )
4739, 46pm2.65da 650 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  -.  q  <  z )
4820, 24, 47nltled 7876 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  <_  q )
49 simplrr 525 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  <  C )
5020, 24, 26, 48, 49lelttrd 7880 . . 3  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  <  C )
5150ralrimiva 2503 . 2  |-  ( (
ph  /\  ( q  e.  U  /\  q  <  C ) )  ->  A. z  e.  L  z  <  C )
5210, 51rexlimddv 2552 1  |-  ( ph  ->  A. z  e.  L  z  <  C )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 697    = wceq 1331    e. wcel 1480   A.wral 2414   E.wrex 2415    i^i cin 3065    C_ wss 3066   (/)c0 3358   class class class wbr 3924  (class class class)co 5767   RRcr 7612    < clt 7793   [,]cicc 9667
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-ltirr 7725  ax-pre-ltwlin 7726  ax-pre-lttrn 7727
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 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-sbc 2905  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-id 4210  df-po 4213  df-iso 4214  df-xp 4540  df-rel 4541  df-cnv 4542  df-co 4543  df-dm 4544  df-iota 5083  df-fun 5120  df-fv 5126  df-ov 5770  df-oprab 5771  df-mpo 5772  df-pnf 7795  df-mnf 7796  df-xr 7797  df-ltxr 7798  df-le 7799  df-icc 9671
This theorem is referenced by:  dedekindicclemub  12763  dedekindicclemloc  12764
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