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

Theorem recexprlemopu 6879
Description: The upper cut of  B is open. Lemma for recexpr 6890. (Contributed by Jim Kingdon, 28-Dec-2019.)
Hypothesis
Ref Expression
recexpr.1  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
Assertion
Ref Expression
recexprlemopu  |-  ( ( A  e.  P.  /\  r  e.  Q.  /\  r  e.  ( 2nd `  B
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
Distinct variable groups:    r, q, x, y, A    B, q,
r, x, y

Proof of Theorem recexprlemopu
StepHypRef Expression
1 recexpr.1 . . . 4  |-  B  = 
<. { x  |  E. y ( x  <Q  y  /\  ( *Q `  y )  e.  ( 2nd `  A ) ) } ,  {
x  |  E. y
( y  <Q  x  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) }
>.
21recexprlemelu 6875 . . 3  |-  ( r  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
3 ltbtwnnqq 6667 . . . . . 6  |-  ( y 
<Q  r  <->  E. q  e.  Q.  ( y  <Q  q  /\  q  <Q  r ) )
43biimpi 118 . . . . 5  |-  ( y 
<Q  r  ->  E. q  e.  Q.  ( y  <Q 
q  /\  q  <Q  r ) )
5 simplr 497 . . . . . . . 8  |-  ( ( ( y  <Q  q  /\  q  <Q  r )  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  q  <Q  r )
6 19.8a 1523 . . . . . . . . . 10  |-  ( ( y  <Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  E. y
( y  <Q  q  /\  ( *Q `  y
)  e.  ( 1st `  A ) ) )
71recexprlemelu 6875 . . . . . . . . . 10  |-  ( q  e.  ( 2nd `  B
)  <->  E. y ( y 
<Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) ) )
86, 7sylibr 132 . . . . . . . . 9  |-  ( ( y  <Q  q  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  q  e.  ( 2nd `  B
) )
98adantlr 461 . . . . . . . 8  |-  ( ( ( y  <Q  q  /\  q  <Q  r )  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  q  e.  ( 2nd `  B ) )
105, 9jca 300 . . . . . . 7  |-  ( ( ( y  <Q  q  /\  q  <Q  r )  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  ( q  <Q  r  /\  q  e.  ( 2nd `  B
) ) )
1110expcom 114 . . . . . 6  |-  ( ( *Q `  y )  e.  ( 1st `  A
)  ->  ( (
y  <Q  q  /\  q  <Q  r )  ->  (
q  <Q  r  /\  q  e.  ( 2nd `  B
) ) ) )
1211reximdv 2463 . . . . 5  |-  ( ( *Q `  y )  e.  ( 1st `  A
)  ->  ( E. q  e.  Q.  (
y  <Q  q  /\  q  <Q  r )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) ) )
134, 12mpan9 275 . . . 4  |-  ( ( y  <Q  r  /\  ( *Q `  y )  e.  ( 1st `  A
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
1413exlimiv 1530 . . 3  |-  ( E. y ( y  <Q 
r  /\  ( *Q `  y )  e.  ( 1st `  A ) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
152, 14sylbi 119 . 2  |-  ( r  e.  ( 2nd `  B
)  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
16153ad2ant3 962 1  |-  ( ( A  e.  P.  /\  r  e.  Q.  /\  r  e.  ( 2nd `  B
) )  ->  E. q  e.  Q.  ( q  <Q 
r  /\  q  e.  ( 2nd `  B ) ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 102    /\ w3a 920    = wceq 1285   E.wex 1422    e. wcel 1434   {cab 2068   E.wrex 2350   <.cop 3409   class class class wbr 3793   ` cfv 4932   1stc1st 5796   2ndc2nd 5797   Q.cnq 6532   *Qcrq 6536    <Q cltq 6537   P.cnp 6543
This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 104  ax-ia2 105  ax-ia3 106  ax-in1 577  ax-in2 578  ax-io 663  ax-5 1377  ax-7 1378  ax-gen 1379  ax-ie1 1423  ax-ie2 1424  ax-8 1436  ax-10 1437  ax-11 1438  ax-i12 1439  ax-bndl 1440  ax-4 1441  ax-13 1445  ax-14 1446  ax-17 1460  ax-i9 1464  ax-ial 1468  ax-i5r 1469  ax-ext 2064  ax-coll 3901  ax-sep 3904  ax-nul 3912  ax-pow 3956  ax-pr 3972  ax-un 4196  ax-setind 4288  ax-iinf 4337
This theorem depends on definitions:  df-bi 115  df-dc 777  df-3or 921  df-3an 922  df-tru 1288  df-fal 1291  df-nf 1391  df-sb 1687  df-eu 1945  df-mo 1946  df-clab 2069  df-cleq 2075  df-clel 2078  df-nfc 2209  df-ne 2247  df-ral 2354  df-rex 2355  df-reu 2356  df-rab 2358  df-v 2604  df-sbc 2817  df-csb 2910  df-dif 2976  df-un 2978  df-in 2980  df-ss 2987  df-nul 3259  df-pw 3392  df-sn 3412  df-pr 3413  df-op 3415  df-uni 3610  df-int 3645  df-iun 3688  df-br 3794  df-opab 3848  df-mpt 3849  df-tr 3884  df-eprel 4052  df-id 4056  df-po 4059  df-iso 4060  df-iord 4129  df-on 4131  df-suc 4134  df-iom 4340  df-xp 4377  df-rel 4378  df-cnv 4379  df-co 4380  df-dm 4381  df-rn 4382  df-res 4383  df-ima 4384  df-iota 4897  df-fun 4934  df-fn 4935  df-f 4936  df-f1 4937  df-fo 4938  df-f1o 4939  df-fv 4940  df-ov 5546  df-oprab 5547  df-mpt2 5548  df-1st 5798  df-2nd 5799  df-recs 5954  df-irdg 6019  df-1o 6065  df-oadd 6069  df-omul 6070  df-er 6172  df-ec 6174  df-qs 6178  df-ni 6556  df-pli 6557  df-mi 6558  df-lti 6559  df-plpq 6596  df-mpq 6597  df-enq 6599  df-nqqs 6600  df-plqqs 6601  df-mqqs 6602  df-1nqqs 6603  df-rq 6604  df-ltnqqs 6605
This theorem is referenced by:  recexprlemrnd  6881
  Copyright terms: Public domain W3C validator