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Theorem suplocexprlemml 7531
Description: Lemma for suplocexpr 7540. The lower 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 ) ) )
Assertion
Ref Expression
suplocexprlemml  |-  ( ph  ->  E. s  e.  Q.  s  e.  U. ( 1st " A ) )
Distinct variable groups:    A, s, x, y    ph, s, x, y
Allowed substitution hints:    ph( z)    A( z)

Proof of Theorem suplocexprlemml
StepHypRef Expression
1 suplocexpr.m . . 3  |-  ( ph  ->  E. x  x  e.  A )
2 suplocexpr.ub . . . . . . 7  |-  ( ph  ->  E. x  e.  P.  A. y  e.  A  y 
<P  x )
3 suplocexpr.loc . . . . . . 7  |-  ( 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 ) ) )
41, 2, 3suplocexprlemss 7530 . . . . . 6  |-  ( ph  ->  A  C_  P. )
54sselda 3097 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  P. )
6 prop 7290 . . . . 5  |-  ( x  e.  P.  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  P. )
7 prml 7292 . . . . 5  |-  ( <.
( 1st `  x
) ,  ( 2nd `  x ) >.  e.  P.  ->  E. s  e.  Q.  s  e.  ( 1st `  x ) )
85, 6, 73syl 17 . . . 4  |-  ( (
ph  /\  x  e.  A )  ->  E. s  e.  Q.  s  e.  ( 1st `  x ) )
98ralrimiva 2505 . . 3  |-  ( ph  ->  A. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )
10 r19.2m 3449 . . 3  |-  ( ( E. x  x  e.  A  /\  A. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )  ->  E. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )
111, 9, 10syl2anc 408 . 2  |-  ( ph  ->  E. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )
12 suplocexprlemell 7528 . . . 4  |-  ( s  e.  U. ( 1st " A )  <->  E. x  e.  A  s  e.  ( 1st `  x ) )
1312rexbii 2442 . . 3  |-  ( E. s  e.  Q.  s  e.  U. ( 1st " A
)  <->  E. s  e.  Q.  E. x  e.  A  s  e.  ( 1st `  x
) )
14 rexcom 2595 . . 3  |-  ( E. s  e.  Q.  E. x  e.  A  s  e.  ( 1st `  x
)  <->  E. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )
1513, 14bitri 183 . 2  |-  ( E. s  e.  Q.  s  e.  U. ( 1st " A
)  <->  E. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )
1611, 15sylibr 133 1  |-  ( ph  ->  E. s  e.  Q.  s  e.  U. ( 1st " A ) )
Colors of variables: wff set class
Syntax hints:    -> wi 4    /\ wa 103    \/ wo 697   E.wex 1468    e. wcel 1480   A.wral 2416   E.wrex 2417   <.cop 3530   U.cuni 3736   class class class wbr 3929   "cima 4542   ` cfv 5123   1stc1st 6036   2ndc2nd 6037   Q.cnq 7095   P.cnp 7106    <P cltp 7110
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 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-coll 4043  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-iinf 4502
This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ral 2421  df-rex 2422  df-reu 2423  df-rab 2425  df-v 2688  df-sbc 2910  df-csb 3004  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-int 3772  df-iun 3815  df-br 3930  df-opab 3990  df-mpt 3991  df-id 4215  df-iom 4505  df-xp 4545  df-rel 4546  df-cnv 4547  df-co 4548  df-dm 4549  df-rn 4550  df-res 4551  df-ima 4552  df-iota 5088  df-fun 5125  df-fn 5126  df-f 5127  df-f1 5128  df-fo 5129  df-f1o 5130  df-fv 5131  df-1st 6038  df-2nd 6039  df-qs 6435  df-ni 7119  df-nqqs 7163  df-inp 7281  df-iltp 7285
This theorem is referenced by:  suplocexprlemex  7537
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