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Theorem dedekindeulemuub 14366
Description: Lemma for dedekindeu 14372. Any element of the upper cut is an upper bound for the lower cut. (Contributed by Jim Kingdon, 2-Feb-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 )
) )
dedekindeulemuub.u  |-  ( ph  ->  A  e.  U )
Assertion
Ref Expression
dedekindeulemuub  |-  ( ph  ->  A. z  e.  L  z  <  A )
Distinct variable groups:    A, q, r, z    L, q, z    U, q, z, r    ph, q,
z
Allowed substitution hints:    ph( r)    L( r)

Proof of Theorem dedekindeulemuub
Dummy variable  a is distinct from all other variables.
StepHypRef Expression
1 dedekindeulemuub.u . . 3  |-  ( ph  ->  A  e.  U )
2 eleq1 2250 . . . . 5  |-  ( r  =  A  ->  (
r  e.  U  <->  A  e.  U ) )
3 breq2 4019 . . . . . 6  |-  ( r  =  A  ->  (
q  <  r  <->  q  <  A ) )
43rexbidv 2488 . . . . 5  |-  ( r  =  A  ->  ( E. q  e.  U  q  <  r  <->  E. q  e.  U  q  <  A ) )
52, 4bibi12d 235 . . . 4  |-  ( r  =  A  ->  (
( r  e.  U  <->  E. q  e.  U  q  <  r )  <->  ( A  e.  U  <->  E. q  e.  U  q  <  A ) ) )
6 dedekindeu.ur . . . 4  |-  ( ph  ->  A. r  e.  RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
7 dedekindeu.uss . . . . 5  |-  ( ph  ->  U  C_  RR )
87, 1sseldd 3168 . . . 4  |-  ( ph  ->  A  e.  RR )
95, 6, 8rspcdva 2858 . . 3  |-  ( ph  ->  ( A  e.  U  <->  E. q  e.  U  q  <  A ) )
101, 9mpbid 147 . 2  |-  ( ph  ->  E. q  e.  U  q  <  A )
11 dedekindeu.lss . . . . . 6  |-  ( ph  ->  L  C_  RR )
1211ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  L  C_  RR )
13 simpr 110 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  z  e.  L )
1412, 13sseldd 3168 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  z  e.  RR )
157ad2antrr 488 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  U  C_  RR )
16 simplrl 535 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  q  e.  U )
1715, 16sseldd 3168 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  q  e.  RR )
188ad2antrr 488 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  A  e.  RR )
19 breq1 4018 . . . . . . . . . 10  |-  ( a  =  q  ->  (
a  <  z  <->  q  <  z ) )
2019rspcev 2853 . . . . . . . . 9  |-  ( ( q  e.  U  /\  q  <  z )  ->  E. a  e.  U  a  <  z )
2116, 20sylan 283 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  E. a  e.  U  a  <  z )
2219cbvrexv 2716 . . . . . . . 8  |-  ( E. a  e.  U  a  <  z  <->  E. q  e.  U  q  <  z )
2321, 22sylib 122 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  E. q  e.  U  q  <  z )
24 eleq1 2250 . . . . . . . . 9  |-  ( r  =  z  ->  (
r  e.  U  <->  z  e.  U ) )
25 breq2 4019 . . . . . . . . . 10  |-  ( r  =  z  ->  (
q  <  r  <->  q  <  z ) )
2625rexbidv 2488 . . . . . . . . 9  |-  ( r  =  z  ->  ( E. q  e.  U  q  <  r  <->  E. q  e.  U  q  <  z ) )
2724, 26bibi12d 235 . . . . . . . 8  |-  ( r  =  z  ->  (
( r  e.  U  <->  E. q  e.  U  q  <  r )  <->  ( z  e.  U  <->  E. q  e.  U  q  <  z ) ) )
286ad3antrrr 492 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  A. r  e.  RR  ( r  e.  U  <->  E. q  e.  U  q  <  r ) )
2914adantr 276 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  z  e.  RR )
3027, 28, 29rspcdva 2858 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  (
z  e.  U  <->  E. q  e.  U  q  <  z ) )
3123, 30mpbird 167 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  z  e.  U )
32 simplll 533 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  ph )
3313adantr 276 . . . . . . 7  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  z  e.  L )
34 dedekindeu.disj . . . . . . . . 9  |-  ( ph  ->  ( L  i^i  U
)  =  (/) )
35 disj 3483 . . . . . . . . 9  |-  ( ( L  i^i  U )  =  (/)  <->  A. z  e.  L  -.  z  e.  U
)
3634, 35sylib 122 . . . . . . . 8  |-  ( ph  ->  A. z  e.  L  -.  z  e.  U
)
3736r19.21bi 2575 . . . . . . 7  |-  ( (
ph  /\  z  e.  L )  ->  -.  z  e.  U )
3832, 33, 37syl2anc 411 . . . . . 6  |-  ( ( ( ( ph  /\  ( q  e.  U  /\  q  <  A ) )  /\  z  e.  L )  /\  q  <  z )  ->  -.  z  e.  U )
3931, 38pm2.65da 662 . . . . 5  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  -.  q  <  z )
4014, 17, 39nltled 8091 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  z  <_  q )
41 simplrr 536 . . . 4  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  q  <  A )
4214, 17, 18, 40, 41lelttrd 8095 . . 3  |-  ( ( ( ph  /\  (
q  e.  U  /\  q  <  A ) )  /\  z  e.  L
)  ->  z  <  A )
4342ralrimiva 2560 . 2  |-  ( (
ph  /\  ( q  e.  U  /\  q  <  A ) )  ->  A. z  e.  L  z  <  A )
4410, 43rexlimddv 2609 1  |-  ( ph  ->  A. z  e.  L  z  <  A )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1363    e. wcel 2158   A.wral 2465   E.wrex 2466    i^i cin 3140    C_ wss 3141   (/)c0 3434   class class class wbr 4015   RRcr 7823    < clt 8005
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 615  ax-in2 616  ax-io 710  ax-5 1457  ax-7 1458  ax-gen 1459  ax-ie1 1503  ax-ie2 1504  ax-8 1514  ax-10 1515  ax-11 1516  ax-i12 1517  ax-bndl 1519  ax-4 1520  ax-17 1536  ax-i9 1540  ax-ial 1544  ax-i5r 1545  ax-13 2160  ax-14 2161  ax-ext 2169  ax-sep 4133  ax-pow 4186  ax-pr 4221  ax-un 4445  ax-setind 4548  ax-cnex 7915  ax-resscn 7916  ax-pre-ltwlin 7937
This theorem depends on definitions:  df-bi 117  df-3an 981  df-tru 1366  df-fal 1369  df-nf 1471  df-sb 1773  df-eu 2039  df-mo 2040  df-clab 2174  df-cleq 2180  df-clel 2183  df-nfc 2318  df-ne 2358  df-nel 2453  df-ral 2470  df-rex 2471  df-rab 2474  df-v 2751  df-dif 3143  df-un 3145  df-in 3147  df-ss 3154  df-nul 3435  df-pw 3589  df-sn 3610  df-pr 3611  df-op 3613  df-uni 3822  df-br 4016  df-opab 4077  df-xp 4644  df-cnv 4646  df-pnf 8007  df-mnf 8008  df-xr 8009  df-ltxr 8010  df-le 8011
This theorem is referenced by:  dedekindeulemub  14367  dedekindeulemloc  14368
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