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Theorem suplocexprlemml 7665
Description: Lemma for suplocexpr 7674. 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 7664 . . . . . 6  |-  ( ph  ->  A  C_  P. )
54sselda 3147 . . . . 5  |-  ( (
ph  /\  x  e.  A )  ->  x  e.  P. )
6 prop 7424 . . . . 5  |-  ( x  e.  P.  ->  <. ( 1st `  x ) ,  ( 2nd `  x
) >.  e.  P. )
7 prml 7426 . . . . 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 2543 . . 3  |-  ( ph  ->  A. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )
10 r19.2m 3500 . . 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 409 . 2  |-  ( ph  ->  E. x  e.  A  E. s  e.  Q.  s  e.  ( 1st `  x ) )
12 suplocexprlemell 7662 . . . 4  |-  ( s  e.  U. ( 1st " A )  <->  E. x  e.  A  s  e.  ( 1st `  x ) )
1312rexbii 2477 . . 3  |-  ( E. s  e.  Q.  s  e.  U. ( 1st " A
)  <->  E. s  e.  Q.  E. x  e.  A  s  e.  ( 1st `  x
) )
14 rexcom 2634 . . 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 703   E.wex 1485    e. wcel 2141   A.wral 2448   E.wrex 2449   <.cop 3584   U.cuni 3794   class class class wbr 3987   "cima 4612   ` cfv 5196   1stc1st 6114   2ndc2nd 6115   Q.cnq 7229   P.cnp 7240    <P cltp 7244
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 4102  ax-sep 4105  ax-pow 4158  ax-pr 4192  ax-un 4416  ax-iinf 4570
This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  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-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-pw 3566  df-sn 3587  df-pr 3588  df-op 3590  df-uni 3795  df-int 3830  df-iun 3873  df-br 3988  df-opab 4049  df-mpt 4050  df-id 4276  df-iom 4573  df-xp 4615  df-rel 4616  df-cnv 4617  df-co 4618  df-dm 4619  df-rn 4620  df-res 4621  df-ima 4622  df-iota 5158  df-fun 5198  df-fn 5199  df-f 5200  df-f1 5201  df-fo 5202  df-f1o 5203  df-fv 5204  df-1st 6116  df-2nd 6117  df-qs 6515  df-ni 7253  df-nqqs 7297  df-inp 7415  df-iltp 7419
This theorem is referenced by:  suplocexprlemex  7671
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