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Theorem suplocexprlemmu 7680
Description: Lemma for suplocexpr 7687. The upper cut of the putative supremum is inhabited. (Contributed by Jim Kingdon, 7-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
suplocexprlemmu  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
Distinct variable groups:    A, s, u, w    x, A, y, s, u    B, s    ph, s, u, x, y
Allowed substitution hints:    ph( z, w)    A( z)    B( x, y, z, w, u)

Proof of Theorem suplocexprlemmu
Dummy variables  j  t are mutually distinct and distinct from all other variables.
StepHypRef Expression
1 suplocexpr.ub . . . 4  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
2 prop 7437 . . . . . . 7  |-  ( x  e.  P.  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  P. )
3 prmu 7440 . . . . . . 7  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  P.  ->  E. s  e.  Q.  s  e.  ( 2nd `  x ) )
42, 3syl 14 . . . . . 6  |-  ( x  e.  P.  ->  E. s  e.  Q.  s  e.  ( 2nd `  x ) )
54ad2antrl 487 . . . . 5  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  E. s  e.  Q.  s  e.  ( 2nd `  x ) )
6 fo2nd 6137 . . . . . . . . . . . . 13  |-  2nd : _V -onto-> _V
7 fofun 5421 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
86, 7ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
9 fvelima 5548 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  t  e.  ( 2nd " A
) )  ->  E. u  e.  A  ( 2nd `  u )  =  t )
108, 9mpan 422 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd " A
)  ->  E. u  e.  A  ( 2nd `  u )  =  t )
1110adantl 275 . . . . . . . . . 10  |-  ( ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  /\  t  e.  ( 2nd " A
) )  ->  E. u  e.  A  ( 2nd `  u )  =  t )
12 suplocexpr.m . . . . . . . . . . . . . . . 16  |-  ( ph  ->  E. x  x  e.  A )
13 suplocexpr.loc . . . . . . . . . . . . . . . 16  |-  ( 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 ) ) )
1412, 1, 13suplocexprlemss 7677 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  C_  P. )
1514ad5antr 493 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  A  C_  P. )
16 simprl 526 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  u  e.  A
)
1715, 16sseldd 3148 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  u  e.  P. )
18 simprl 526 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  x  e.  P. )
1918ad4antr 491 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  x  e.  P. )
20 breq1 3992 . . . . . . . . . . . . . . 15  |-  ( y  =  u  ->  (
y  <P  x  <->  u  <P  x ) )
21 simprr 527 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  A. y  e.  A  y  <P  x )
2221ad4antr 491 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  A. y  e.  A  y  <P  x )
2320, 22, 16rspcdva 2839 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  u  <P  x
)
24 ltsopr 7558 . . . . . . . . . . . . . . . . 17  |-  <P  Or  P.
25 so2nr 4306 . . . . . . . . . . . . . . . . 17  |-  ( ( 
<P  Or  P.  /\  (
u  e.  P.  /\  x  e.  P. )
)  ->  -.  (
u  <P  x  /\  x  <P  u ) )
2624, 25mpan 422 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  P.  /\  x  e.  P. )  ->  -.  ( u  <P  x  /\  x  <P  u
) )
2717, 19, 26syl2anc 409 . . . . . . . . . . . . . . 15  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  -.  ( u  <P  x  /\  x  <P  u ) )
28 imnan 685 . . . . . . . . . . . . . . 15  |-  ( ( u  <P  x  ->  -.  x  <P  u )  <->  -.  ( u  <P  x  /\  x  <P  u ) )
2927, 28sylibr 133 . . . . . . . . . . . . . 14  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  ( u  <P  x  ->  -.  x  <P  u ) )
3023, 29mpd 13 . . . . . . . . . . . . 13  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  -.  x  <P  u )
31 aptiprlemu 7602 . . . . . . . . . . . . 13  |-  ( ( u  e.  P.  /\  x  e.  P.  /\  -.  x  <P  u )  -> 
( 2nd `  x
)  C_  ( 2nd `  u ) )
3217, 19, 30, 31syl3anc 1233 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  ( 2nd `  x
)  C_  ( 2nd `  u ) )
33 simpllr 529 . . . . . . . . . . . 12  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  s  e.  ( 2nd `  x ) )
3432, 33sseldd 3148 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  s  e.  ( 2nd `  u ) )
35 simprr 527 . . . . . . . . . . 11  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  ( 2nd `  u
)  =  t )
3634, 35eleqtrd 2249 . . . . . . . . . 10  |-  ( ( ( ( ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x
) )  /\  t  e.  ( 2nd " A
) )  /\  (
u  e.  A  /\  ( 2nd `  u )  =  t ) )  ->  s  e.  t )
3711, 36rexlimddv 2592 . . . . . . . . 9  |-  ( ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  /\  t  e.  ( 2nd " A
) )  ->  s  e.  t )
3837ralrimiva 2543 . . . . . . . 8  |-  ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  ->  A. t  e.  ( 2nd " A
) s  e.  t )
39 vex 2733 . . . . . . . . 9  |-  s  e. 
_V
4039elint2 3838 . . . . . . . 8  |-  ( s  e.  |^| ( 2nd " A
)  <->  A. t  e.  ( 2nd " A ) s  e.  t )
4138, 40sylibr 133 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  ->  s  e.  |^| ( 2nd " A
) )
4241ex 114 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P.  /\  A. y  e.  A  y 
<P  x ) )  /\  s  e.  Q. )  ->  ( s  e.  ( 2nd `  x )  ->  s  e.  |^| ( 2nd " A ) ) )
4342reximdva 2572 . . . . 5  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  ( E. s  e.  Q.  s  e.  ( 2nd `  x )  ->  E. s  e.  Q.  s  e.  |^| ( 2nd " A ) ) )
445, 43mpd 13 . . . 4  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  E. s  e.  Q.  s  e.  |^| ( 2nd " A ) )
451, 44rexlimddv 2592 . . 3  |-  ( ph  ->  E. s  e.  Q.  s  e.  |^| ( 2nd " A ) )
46 simprr 527 . . . . . . 7  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  e.  |^| ( 2nd " A ) )
47 simprl 526 . . . . . . . . 9  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  e.  Q. )
48 1nq 7328 . . . . . . . . 9  |-  1Q  e.  Q.
49 addclnq 7337 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  1Q  e.  Q. )  -> 
( s  +Q  1Q )  e.  Q. )
5047, 48, 49sylancl 411 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
( s  +Q  1Q )  e.  Q. )
51 ltaddnq 7369 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  1Q  e.  Q. )  -> 
s  <Q  ( s  +Q  1Q ) )
5247, 48, 51sylancl 411 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  <Q  ( s  +Q  1Q ) )
53 breq2 3993 . . . . . . . . 9  |-  ( j  =  ( s  +Q  1Q )  ->  (
s  <Q  j  <->  s  <Q  ( s  +Q  1Q ) ) )
5453rspcev 2834 . . . . . . . 8  |-  ( ( ( s  +Q  1Q )  e.  Q.  /\  s  <Q  ( s  +Q  1Q ) )  ->  E. j  e.  Q.  s  <Q  j
)
5550, 52, 54syl2anc 409 . . . . . . 7  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  s  <Q  j )
56 breq1 3992 . . . . . . . . 9  |-  ( w  =  s  ->  (
w  <Q  j  <->  s  <Q  j ) )
5756rexbidv 2471 . . . . . . . 8  |-  ( w  =  s  ->  ( E. j  e.  Q.  w  <Q  j  <->  E. j  e.  Q.  s  <Q  j
) )
5857rspcev 2834 . . . . . . 7  |-  ( ( s  e.  |^| ( 2nd " A )  /\  E. j  e.  Q.  s  <Q  j )  ->  E. w  e.  |^| ( 2nd " A
) E. j  e. 
Q.  w  <Q  j
)
5946, 55, 58syl2anc 409 . . . . . 6  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. w  e.  |^| ( 2nd " A ) E. j  e.  Q.  w  <Q  j )
60 rexcom 2634 . . . . . 6  |-  ( E. w  e.  |^| ( 2nd " A ) E. j  e.  Q.  w  <Q  j  <->  E. j  e.  Q.  E. w  e.  |^| ( 2nd " A ) w 
<Q  j )
6159, 60sylib 121 . . . . 5  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  E. w  e.  |^| ( 2nd " A ) w 
<Q  j )
62 ssid 3167 . . . . . 6  |-  Q.  C_  Q.
63 rexss 3214 . . . . . 6  |-  ( Q.  C_  Q.  ->  ( E. j  e.  Q.  E. w  e.  |^| ( 2nd " A
) w  <Q  j  <->  E. j  e.  Q.  (
j  e.  Q.  /\  E. w  e.  |^| ( 2nd " A ) w 
<Q  j ) ) )
6462, 63ax-mp 5 . . . . 5  |-  ( E. j  e.  Q.  E. w  e.  |^| ( 2nd " A ) w  <Q  j  <->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
6561, 64sylib 121 . . . 4  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
66 suplocexpr.b . . . . . . . . . 10  |-  B  = 
<. U. ( 1st " A
) ,  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } >.
6766suplocexprlem2b 7676 . . . . . . . . 9  |-  ( A 
C_  P.  ->  ( 2nd `  B )  =  {
u  e.  Q.  |  E. w  e.  |^| ( 2nd " A ) w 
<Q  u } )
6814, 67syl 14 . . . . . . . 8  |-  ( ph  ->  ( 2nd `  B
)  =  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } )
6968eleq2d 2240 . . . . . . 7  |-  ( ph  ->  ( j  e.  ( 2nd `  B )  <-> 
j  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
70 breq2 3993 . . . . . . . . 9  |-  ( u  =  j  ->  (
w  <Q  u  <->  w  <Q  j ) )
7170rexbidv 2471 . . . . . . . 8  |-  ( u  =  j  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
7271elrab 2886 . . . . . . 7  |-  ( j  e.  { u  e. 
Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } 
<->  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
7369, 72bitrdi 195 . . . . . 6  |-  ( ph  ->  ( j  e.  ( 2nd `  B )  <-> 
( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
) )
7473rexbidv 2471 . . . . 5  |-  ( ph  ->  ( E. j  e. 
Q.  j  e.  ( 2nd `  B )  <->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
) )
7574adantr 274 . . . 4  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
( E. j  e. 
Q.  j  e.  ( 2nd `  B )  <->  E. j  e.  Q.  ( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
) )
7665, 75mpbird 166 . . 3  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  j  e.  ( 2nd `  B ) )
7745, 76rexlimddv 2592 . 2  |-  ( ph  ->  E. j  e.  Q.  j  e.  ( 2nd `  B ) )
78 eleq1w 2231 . . 3  |-  ( j  =  s  ->  (
j  e.  ( 2nd `  B )  <->  s  e.  ( 2nd `  B ) ) )
7978cbvrexv 2697 . 2  |-  ( E. j  e.  Q.  j  e.  ( 2nd `  B
)  <->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
8077, 79sylib 121 1  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 103    <-> wb 104    \/ wo 703    = wceq 1348   E.wex 1485    e. wcel 2141   A.wral 2448   E.wrex 2449   {crab 2452   _Vcvv 2730    C_ wss 3121   <.cop 3586   U.cuni 3796   |^|cint 3831   class class class wbr 3989    Or wor 4280   "cima 4614   Fun wfun 5192   -onto->wfo 5196   ` cfv 5198  (class class class)co 5853   1stc1st 6117   2ndc2nd 6118   Q.cnq 7242   1Qc1q 7243    +Q cplq 7244    <Q cltq 7247   P.cnp 7253    <P cltp 7257
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-coll 4104  ax-sep 4107  ax-nul 4115  ax-pow 4160  ax-pr 4194  ax-un 4418  ax-setind 4521  ax-iinf 4572
This theorem depends on definitions:  df-bi 116  df-dc 830  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-ral 2453  df-rex 2454  df-reu 2455  df-rab 2457  df-v 2732  df-sbc 2956  df-csb 3050  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-int 3832  df-iun 3875  df-br 3990  df-opab 4051  df-mpt 4052  df-tr 4088  df-eprel 4274  df-id 4278  df-po 4281  df-iso 4282  df-iord 4351  df-on 4353  df-suc 4356  df-iom 4575  df-xp 4617  df-rel 4618  df-cnv 4619  df-co 4620  df-dm 4621  df-rn 4622  df-res 4623  df-ima 4624  df-iota 5160  df-fun 5200  df-fn 5201  df-f 5202  df-f1 5203  df-fo 5204  df-f1o 5205  df-fv 5206  df-ov 5856  df-oprab 5857  df-mpo 5858  df-1st 6119  df-2nd 6120  df-recs 6284  df-irdg 6349  df-1o 6395  df-2o 6396  df-oadd 6399  df-omul 6400  df-er 6513  df-ec 6515  df-qs 6519  df-ni 7266  df-pli 7267  df-mi 7268  df-lti 7269  df-plpq 7306  df-mpq 7307  df-enq 7309  df-nqqs 7310  df-plqqs 7311  df-mqqs 7312  df-1nqqs 7313  df-rq 7314  df-ltnqqs 7315  df-enq0 7386  df-nq0 7387  df-0nq0 7388  df-plq0 7389  df-mq0 7390  df-inp 7428  df-iltp 7432
This theorem is referenced by:  suplocexprlemex  7684
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