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Theorem suplocexprlemmu 7708
Description: Lemma for suplocexpr 7715. 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 7465 . . . . . . 7  |-  ( x  e.  P.  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  P. )
3 prmu 7468 . . . . . . 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 490 . . . . 5  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  E. s  e.  Q.  s  e.  ( 2nd `  x ) )
6 fo2nd 6153 . . . . . . . . . . . . 13  |-  2nd : _V -onto-> _V
7 fofun 5435 . . . . . . . . . . . . 13  |-  ( 2nd
: _V -onto-> _V  ->  Fun 
2nd )
86, 7ax-mp 5 . . . . . . . . . . . 12  |-  Fun  2nd
9 fvelima 5563 . . . . . . . . . . . 12  |-  ( ( Fun  2nd  /\  t  e.  ( 2nd " A
) )  ->  E. u  e.  A  ( 2nd `  u )  =  t )
108, 9mpan 424 . . . . . . . . . . 11  |-  ( t  e.  ( 2nd " A
)  ->  E. u  e.  A  ( 2nd `  u )  =  t )
1110adantl 277 . . . . . . . . . 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 7705 . . . . . . . . . . . . . . 15  |-  ( ph  ->  A  C_  P. )
1514ad5antr 496 . . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . 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 3156 . . . . . . . . . . . . 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 529 . . . . . . . . . . . . . 14  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  x  e.  P. )
1918ad4antr 494 . . . . . . . . . . . . 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 4003 . . . . . . . . . . . . . . 15  |-  ( y  =  u  ->  (
y  <P  x  <->  u  <P  x ) )
21 simprr 531 . . . . . . . . . . . . . . . 16  |-  ( (
ph  /\  ( x  e.  P.  /\  A. y  e.  A  y  <P  x ) )  ->  A. y  e.  A  y  <P  x )
2221ad4antr 494 . . . . . . . . . . . . . . 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 2846 . . . . . . . . . . . . . 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 7586 . . . . . . . . . . . . . . . . 17  |-  <P  Or  P.
25 so2nr 4318 . . . . . . . . . . . . . . . . 17  |-  ( ( 
<P  Or  P.  /\  (
u  e.  P.  /\  x  e.  P. )
)  ->  -.  (
u  <P  x  /\  x  <P  u ) )
2624, 25mpan 424 . . . . . . . . . . . . . . . 16  |-  ( ( u  e.  P.  /\  x  e.  P. )  ->  -.  ( u  <P  x  /\  x  <P  u
) )
2717, 19, 26syl2anc 411 . . . . . . . . . . . . . . 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 690 . . . . . . . . . . . . . . 15  |-  ( ( u  <P  x  ->  -.  x  <P  u )  <->  -.  ( u  <P  x  /\  x  <P  u ) )
2927, 28sylibr 134 . . . . . . . . . . . . . 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 7630 . . . . . . . . . . . . 13  |-  ( ( u  e.  P.  /\  x  e.  P.  /\  -.  x  <P  u )  -> 
( 2nd `  x
)  C_  ( 2nd `  u ) )
3217, 19, 30, 31syl3anc 1238 . . . . . . . . . . . 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 534 . . . . . . . . . . . 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 3156 . . . . . . . . . . 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 531 . . . . . . . . . . 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 2256 . . . . . . . . . 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 2599 . . . . . . . . 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 2550 . . . . . . . 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 2740 . . . . . . . . 9  |-  s  e. 
_V
4039elint2 3849 . . . . . . . 8  |-  ( s  e.  |^| ( 2nd " A
)  <->  A. t  e.  ( 2nd " A ) s  e.  t )
4138, 40sylibr 134 . . . . . . 7  |-  ( ( ( ( ph  /\  ( x  e.  P.  /\ 
A. y  e.  A  y  <P  x ) )  /\  s  e.  Q. )  /\  s  e.  ( 2nd `  x ) )  ->  s  e.  |^| ( 2nd " A
) )
4241ex 115 . . . . . 6  |-  ( ( ( ph  /\  (
x  e.  P.  /\  A. y  e.  A  y 
<P  x ) )  /\  s  e.  Q. )  ->  ( s  e.  ( 2nd `  x )  ->  s  e.  |^| ( 2nd " A ) ) )
4342reximdva 2579 . . . . 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 2599 . . 3  |-  ( ph  ->  E. s  e.  Q.  s  e.  |^| ( 2nd " A ) )
46 simprr 531 . . . . . . 7  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  e.  |^| ( 2nd " A ) )
47 simprl 529 . . . . . . . . 9  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  e.  Q. )
48 1nq 7356 . . . . . . . . 9  |-  1Q  e.  Q.
49 addclnq 7365 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  1Q  e.  Q. )  -> 
( s  +Q  1Q )  e.  Q. )
5047, 48, 49sylancl 413 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
( s  +Q  1Q )  e.  Q. )
51 ltaddnq 7397 . . . . . . . . 9  |-  ( ( s  e.  Q.  /\  1Q  e.  Q. )  -> 
s  <Q  ( s  +Q  1Q ) )
5247, 48, 51sylancl 413 . . . . . . . 8  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  -> 
s  <Q  ( s  +Q  1Q ) )
53 breq2 4004 . . . . . . . . 9  |-  ( j  =  ( s  +Q  1Q )  ->  (
s  <Q  j  <->  s  <Q  ( s  +Q  1Q ) ) )
5453rspcev 2841 . . . . . . . 8  |-  ( ( ( s  +Q  1Q )  e.  Q.  /\  s  <Q  ( s  +Q  1Q ) )  ->  E. j  e.  Q.  s  <Q  j
)
5550, 52, 54syl2anc 411 . . . . . . 7  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  s  <Q  j )
56 breq1 4003 . . . . . . . . 9  |-  ( w  =  s  ->  (
w  <Q  j  <->  s  <Q  j ) )
5756rexbidv 2478 . . . . . . . 8  |-  ( w  =  s  ->  ( E. j  e.  Q.  w  <Q  j  <->  E. j  e.  Q.  s  <Q  j
) )
5857rspcev 2841 . . . . . . 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 411 . . . . . 6  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. w  e.  |^| ( 2nd " A ) E. j  e.  Q.  w  <Q  j )
60 rexcom 2641 . . . . . 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 122 . . . . 5  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  E. w  e.  |^| ( 2nd " A ) w 
<Q  j )
62 ssid 3175 . . . . . 6  |-  Q.  C_  Q.
63 rexss 3222 . . . . . 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 122 . . . 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 7704 . . . . . . . . 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 2247 . . . . . . 7  |-  ( ph  ->  ( j  e.  ( 2nd `  B )  <-> 
j  e.  { u  e.  Q.  |  E. w  e.  |^| ( 2nd " A
) w  <Q  u } ) )
70 breq2 4004 . . . . . . . . 9  |-  ( u  =  j  ->  (
w  <Q  u  <->  w  <Q  j ) )
7170rexbidv 2478 . . . . . . . 8  |-  ( u  =  j  ->  ( E. w  e.  |^| ( 2nd " A ) w 
<Q  u  <->  E. w  e.  |^| ( 2nd " A ) w  <Q  j )
)
7271elrab 2893 . . . . . . 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 196 . . . . . 6  |-  ( ph  ->  ( j  e.  ( 2nd `  B )  <-> 
( j  e.  Q.  /\ 
E. w  e.  |^| ( 2nd " A ) w  <Q  j )
) )
7473rexbidv 2478 . . . . 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 276 . . . 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 167 . . 3  |-  ( (
ph  /\  ( s  e.  Q.  /\  s  e. 
|^| ( 2nd " A
) ) )  ->  E. j  e.  Q.  j  e.  ( 2nd `  B ) )
7745, 76rexlimddv 2599 . 2  |-  ( ph  ->  E. j  e.  Q.  j  e.  ( 2nd `  B ) )
78 eleq1w 2238 . . 3  |-  ( j  =  s  ->  (
j  e.  ( 2nd `  B )  <->  s  e.  ( 2nd `  B ) ) )
7978cbvrexv 2704 . 2  |-  ( E. j  e.  Q.  j  e.  ( 2nd `  B
)  <->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
8077, 79sylib 122 1  |-  ( ph  ->  E. s  e.  Q.  s  e.  ( 2nd `  B ) )
Colors of variables: wff set class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 104    <-> wb 105    \/ wo 708    = wceq 1353   E.wex 1492    e. wcel 2148   A.wral 2455   E.wrex 2456   {crab 2459   _Vcvv 2737    C_ wss 3129   <.cop 3594   U.cuni 3807   |^|cint 3842   class class class wbr 4000    Or wor 4292   "cima 4626   Fun wfun 5206   -onto->wfo 5210   ` cfv 5212  (class class class)co 5869   1stc1st 6133   2ndc2nd 6134   Q.cnq 7270   1Qc1q 7271    +Q cplq 7272    <Q cltq 7275   P.cnp 7281    <P cltp 7285
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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-coll 4115  ax-sep 4118  ax-nul 4126  ax-pow 4171  ax-pr 4206  ax-un 4430  ax-setind 4533  ax-iinf 4584
This theorem depends on definitions:  df-bi 117  df-dc 835  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-ral 2460  df-rex 2461  df-reu 2462  df-rab 2464  df-v 2739  df-sbc 2963  df-csb 3058  df-dif 3131  df-un 3133  df-in 3135  df-ss 3142  df-nul 3423  df-pw 3576  df-sn 3597  df-pr 3598  df-op 3600  df-uni 3808  df-int 3843  df-iun 3886  df-br 4001  df-opab 4062  df-mpt 4063  df-tr 4099  df-eprel 4286  df-id 4290  df-po 4293  df-iso 4294  df-iord 4363  df-on 4365  df-suc 4368  df-iom 4587  df-xp 4629  df-rel 4630  df-cnv 4631  df-co 4632  df-dm 4633  df-rn 4634  df-res 4635  df-ima 4636  df-iota 5174  df-fun 5214  df-fn 5215  df-f 5216  df-f1 5217  df-fo 5218  df-f1o 5219  df-fv 5220  df-ov 5872  df-oprab 5873  df-mpo 5874  df-1st 6135  df-2nd 6136  df-recs 6300  df-irdg 6365  df-1o 6411  df-2o 6412  df-oadd 6415  df-omul 6416  df-er 6529  df-ec 6531  df-qs 6535  df-ni 7294  df-pli 7295  df-mi 7296  df-lti 7297  df-plpq 7334  df-mpq 7335  df-enq 7337  df-nqqs 7338  df-plqqs 7339  df-mqqs 7340  df-1nqqs 7341  df-rq 7342  df-ltnqqs 7343  df-enq0 7414  df-nq0 7415  df-0nq0 7416  df-plq0 7417  df-mq0 7418  df-inp 7456  df-iltp 7460
This theorem is referenced by:  suplocexprlemex  7712
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