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Theorem suplocexprlemdisj 7780
Description: Lemma for suplocexpr 7785. The putative supremum is disjoint. (Contributed by Jim Kingdon, 9-Jan-2024.)
Hypotheses
Ref Expression
suplocexpr.m  |-  ( ph  ->  E. x  x  e.  A )
suplocexpr.ub  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
suplocexpr.loc  |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  (
x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )
suplocexpr.b  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
Assertion
Ref Expression
suplocexprlemdisj  |-  ( ph  ->  A. q  e.  Q.  -.  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )
Distinct variable groups:    w, A, u   
x, A, y    w, B    ph, q, w    ph, x, y    u, q
Allowed substitution hints:    ph( z, u)    A( z, q)    B( x, y, z, u, q)

Proof of Theorem suplocexprlemdisj
Dummy variables  s  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 simprl 529 . . . . 5  |-  ( ( ( ph  /\  q  e.  Q. )  /\  (
q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  ->  q  e.  U. ( 1st " A ) )
2 suplocexprlemell 7773 . . . . 5  |-  ( q  e.  U. ( 1st " A )  <->  E. s  e.  A  q  e.  ( 1st `  s ) )
31, 2sylib 122 . . . 4  |-  ( ( ( ph  /\  q  e.  Q. )  /\  (
q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  ->  E. s  e.  A  q  e.  ( 1st `  s ) )
4 simprr 531 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  q  e.  ( 1st `  s ) )
5 simplrr 536 . . . . . . . . 9  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  q  e.  ( 2nd `  B ) )
6 suplocexpr.m . . . . . . . . . . . . 13  |-  ( ph  ->  E. x  x  e.  A )
7 suplocexpr.ub . . . . . . . . . . . . 13  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
8 suplocexpr.loc . . . . . . . . . . . . 13  |-  ( ph  ->  A. x  e.  P.  A. y  e.  P.  (
x  <P  y  ->  ( E. z  e.  A  x  <P  z  \/  A. z  e.  A  z  <P  y ) ) )
96, 7, 8suplocexprlemss 7775 . . . . . . . . . . . 12  |-  ( ph  ->  A  C_  P. )
109ad3antrrr 492 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  A  C_  P. )
11 suplocexpr.b . . . . . . . . . . . . 13  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
1211suplocexprlem2b 7774 . . . . . . . . . . . 12  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
1312eleq2d 2263 . . . . . . . . . . 11  |-  ( A 
C_  P.  ->  ( q  e.  ( 2nd `  B
)  <->  q  e.  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } ) )
1410, 13syl 14 . . . . . . . . . 10  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  ( q  e.  ( 2nd `  B
)  <->  q  e.  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } ) )
15 breq2 4033 . . . . . . . . . . . 12  |-  ( u  =  q  ->  (
w  <Q  u  <->  w  <Q  q ) )
1615rexbidv 2495 . . . . . . . . . . 11  |-  ( u  =  q  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  q )
)
1716elrab 2916 . . . . . . . . . 10  |-  ( q  e.  { u  e. 
Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } 
<->  ( q  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  q )
)
1814, 17bitrdi 196 . . . . . . . . 9  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  ( q  e.  ( 2nd `  B
)  <->  ( q  e. 
Q.  /\  E. w  e.  |^| ( 2nd " A
) w  <Q  q
) ) )
195, 18mpbid 147 . . . . . . . 8  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  ( q  e. 
Q.  /\  E. w  e.  |^| ( 2nd " A
) w  <Q  q
) )
2019simprd 114 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  E. w  e.  |^| ( 2nd " A ) w  <Q  q )
21 simprr 531 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  w  <Q  q )
2210adantr 276 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  A  C_ 
P. )
23 simplrl 535 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  s  e.  A )
2422, 23sseldd 3180 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  s  e.  P. )
25 prop 7535 . . . . . . . . . 10  |-  ( s  e.  P.  ->  <. ( 1st `  s ) ,  ( 2nd `  s
) >.  e.  P. )
2624, 25syl 14 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  <. ( 1st `  s ) ,  ( 2nd `  s
) >.  e.  P. )
27 eleq2 2257 . . . . . . . . . 10  |-  ( t  =  ( 2nd `  s
)  ->  ( w  e.  t  <->  w  e.  ( 2nd `  s ) ) )
28 simprl 529 . . . . . . . . . . 11  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  w  e.  |^| ( 2nd " A
) )
29 vex 2763 . . . . . . . . . . . 12  |-  w  e. 
_V
3029elint2 3877 . . . . . . . . . . 11  |-  ( w  e.  |^| ( 2nd " A
)  <->  A. t  e.  ( 2nd " A ) w  e.  t )
3128, 30sylib 122 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  A. t  e.  ( 2nd " A
) w  e.  t )
32 fo2nd 6211 . . . . . . . . . . . . . 14  |-  2nd : _V -onto-> _V
33 fofun 5477 . . . . . . . . . . . . . 14  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
3432, 33ax-mp 5 . . . . . . . . . . . . 13  |-  Fun  2nd
35 vex 2763 . . . . . . . . . . . . . 14  |-  s  e. 
_V
36 fof 5476 . . . . . . . . . . . . . . . 16  |-  ( 2nd
: _V -onto-> _V  ->  2nd
: _V --> _V )
3732, 36ax-mp 5 . . . . . . . . . . . . . . 15  |-  2nd : _V
--> _V
3837fdmi 5411 . . . . . . . . . . . . . 14  |-  dom  2nd  =  _V
3935, 38eleqtrri 2269 . . . . . . . . . . . . 13  |-  s  e. 
dom  2nd
40 funfvima 5790 . . . . . . . . . . . . 13  |-  ( ( Fun  2nd  /\  s  e.  dom  2nd )  -> 
( s  e.  A  ->  ( 2nd `  s
)  e.  ( 2nd " A ) ) )
4134, 39, 40mp2an 426 . . . . . . . . . . . 12  |-  ( s  e.  A  ->  ( 2nd `  s )  e.  ( 2nd " A
) )
4241ad2antrl 490 . . . . . . . . . . 11  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  ( 2nd `  s
)  e.  ( 2nd " A ) )
4342adantr 276 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  ( 2nd `  s )  e.  ( 2nd " A
) )
4427, 31, 43rspcdva 2869 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  w  e.  ( 2nd `  s
) )
45 prcunqu 7545 . . . . . . . . 9  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  w  e.  ( 2nd `  s ) )  -> 
( w  <Q  q  ->  q  e.  ( 2nd `  s ) ) )
4626, 44, 45syl2anc 411 . . . . . . . 8  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  (
w  <Q  q  ->  q  e.  ( 2nd `  s
) ) )
4721, 46mpd 13 . . . . . . 7  |-  ( ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  /\  ( w  e. 
|^| ( 2nd " A
)  /\  w  <Q  q ) )  ->  q  e.  ( 2nd `  s
) )
4820, 47rexlimddv 2616 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  q  e.  ( 2nd `  s ) )
494, 48jca 306 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
50 simprl 529 . . . . . . . 8  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  s  e.  A
)
5110, 50sseldd 3180 . . . . . . 7  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  s  e.  P. )
5251, 25syl 14 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  <. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P. )
53 simpllr 534 . . . . . 6  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  q  e.  Q. )
54 prdisj 7552 . . . . . 6  |-  ( (
<. ( 1st `  s
) ,  ( 2nd `  s ) >.  e.  P.  /\  q  e.  Q. )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
5552, 53, 54syl2anc 411 . . . . 5  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  ->  -.  ( q  e.  ( 1st `  s
)  /\  q  e.  ( 2nd `  s ) ) )
5649, 55pm2.21fal 1384 . . . 4  |-  ( ( ( ( ph  /\  q  e.  Q. )  /\  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  /\  (
s  e.  A  /\  q  e.  ( 1st `  s ) ) )  -> F.  )
573, 56rexlimddv 2616 . . 3  |-  ( ( ( ph  /\  q  e.  Q. )  /\  (
q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )  -> F.  )
5857inegd 1383 . 2  |-  ( (
ph  /\  q  e.  Q. )  ->  -.  (
q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )
5958ralrimiva 2567 1  |-  ( ph  ->  A. q  e.  Q.  -.  ( q  e.  U. ( 1st " A )  /\  q  e.  ( 2nd `  B ) ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 709    = wceq 1364   F. wfal 1369   E.wex 1503    e. wcel 2164   A.wral 2472   E.wrex 2473   {crab 2476   _Vcvv 2760    C_ wss 3153   <.cop 3621   U.cuni 3835   |^|cint 3870   class class class wbr 4029   dom cdm 4659   "cima 4662   Fun wfun 5248   -->wf 5250   -onto->wfo 5252   ` cfv 5254   1stc1st 6191   2ndc2nd 6192   Q.cnq 7340    <Q cltq 7345   P.cnp 7351    <P cltp 7355
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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-coll 4144  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-iinf 4620
This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-reu 2479  df-rab 2481  df-v 2762  df-sbc 2986  df-csb 3081  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-int 3871  df-iun 3914  df-br 4030  df-opab 4091  df-mpt 4092  df-id 4324  df-iom 4623  df-xp 4665  df-rel 4666  df-cnv 4667  df-co 4668  df-dm 4669  df-rn 4670  df-res 4671  df-ima 4672  df-iota 5215  df-fun 5256  df-fn 5257  df-f 5258  df-f1 5259  df-fo 5260  df-f1o 5261  df-fv 5262  df-1st 6193  df-2nd 6194  df-qs 6593  df-ni 7364  df-nqqs 7408  df-ltnqqs 7413  df-inp 7526  df-iltp 7530
This theorem is referenced by:  suplocexprlemex  7782
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