ILE Home Intuitionistic Logic Explorer < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  ILE Home  >  Th. List  >  dedekindicclemuub Unicode version

Theorem dedekindicclemuub 13398
Description: Lemma for dedekindicc 13405. 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 2233 . . . . 5  |-  ( r  =  C  ->  (
r  e.  U  <->  C  e.  U ) )
3 breq2 3993 . . . . . 6  |-  ( r  =  C  ->  (
q  <  r  <->  q  <  C ) )
43rexbidv 2471 . . . . 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 3148 . . . 4  |-  ( ph  ->  C  e.  ( A [,] B ) )
95, 6, 8rspcdva 2839 . . 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 9912 . . . . . . 7  |-  ( ( A  e.  RR  /\  B  e.  RR )  ->  ( A [,] B
)  C_  RR )
1411, 12, 13syl2anc 409 . . . . . 6  |-  ( ph  ->  ( A [,] B
)  C_  RR )
1514ad2antrr 485 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  ( A [,] B )  C_  RR )
16 dedekindicc.lss . . . . . . 7  |-  ( ph  ->  L  C_  ( A [,] B ) )
1716ad2antrr 485 . . . . . 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 3148 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  e.  ( A [,] B ) )
2015, 19sseldd 3148 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  e.  RR )
217ad2antrr 485 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  U  C_  ( A [,] B ) )
22 simplrl 530 . . . . . 6  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  e.  U )
2321, 22sseldd 3148 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  e.  ( A [,] B ) )
2415, 23sseldd 3148 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  e.  RR )
2514, 8sseldd 3148 . . . . 5  |-  ( ph  ->  C  e.  RR )
2625ad2antrr 485 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  C  e.  RR )
27 breq1 3992 . . . . . . . . . 10  |-  ( a  =  q  ->  (
a  <  z  <->  q  <  z ) )
2827rspcev 2834 . . . . . . . . 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 2697 . . . . . . . 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 2233 . . . . . . . . 9  |-  ( r  =  z  ->  (
r  e.  U  <->  z  e.  U ) )
33 breq2 3993 . . . . . . . . . 10  |-  ( r  =  z  ->  (
q  <  r  <->  q  <  z ) )
3433rexbidv 2471 . . . . . . . . 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 489 . . . . . . . 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 2839 . . . . . . 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 528 . . . . . . 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 3463 . . . . . . . . 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 2558 . . . . . . 7  |-  ( (
ph  /\  z  e.  L )  ->  -.  z  e.  U )
4640, 41, 45syl2anc 409 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  C ) )  /\  z  e.  L )  /\  q  <  z )  ->  -.  z  e.  U )
4739, 46pm2.65da 656 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  -.  q  <  z )
4820, 24, 47nltled 8040 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  <_  q )
49 simplrr 531 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  q  <  C )
5020, 24, 26, 48, 49lelttrd 8044 . . 3  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  C ) )  /\  z  e.  L
)  ->  z  <  C )
5150ralrimiva 2543 . 2  |-  ( (
ph  /\  ( q  e.  U  /\  q  <  C ) )  ->  A. z  e.  L  z  <  C )
5210, 51rexlimddv 2592 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 703    = wceq 1348    e. wcel 2141   A.wral 2448   E.wrex 2449    i^i cin 3120    C_ wss 3121   (/)c0 3414   class class class wbr 3989  (class class class)co 5853   RRcr 7773    < clt 7954   [,]cicc 9848
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 609  ax-in2 610  ax-io 704  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-10 1498  ax-11 1499  ax-i12 1500  ax-bndl 1502  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-13 2143  ax-14 2144  ax-ext 2152  ax-sep 4107  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-cnex 7865  ax-resscn 7866  ax-pre-ltirr 7886  ax-pre-ltwlin 7887  ax-pre-lttrn 7888
This theorem depends on definitions:  df-bi 116  df-3or 974  df-3an 975  df-tru 1351  df-fal 1354  df-nf 1454  df-sb 1756  df-eu 2022  df-mo 2023  df-clab 2157  df-cleq 2163  df-clel 2166  df-nfc 2301  df-ne 2341  df-nel 2436  df-ral 2453  df-rex 2454  df-rab 2457  df-v 2732  df-sbc 2956  df-dif 3123  df-un 3125  df-in 3127  df-ss 3134  df-nul 3415  df-pw 3568  df-sn 3589  df-pr 3590  df-op 3592  df-uni 3797  df-br 3990  df-opab 4051  df-id 4278  df-po 4281  df-iso 4282  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-iota 5160  df-fun 5200  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-pnf 7956  df-mnf 7957  df-xr 7958  df-ltxr 7959  df-le 7960  df-icc 9852
This theorem is referenced by:  dedekindicclemub  13399  dedekindicclemloc  13400
  Copyright terms: Public domain W3C validator