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Theorem supfirege 10526
Description: The supremum of a finite set of real numbers is greater than or equal to all the real numbers of the set. (Contributed by AV, 1-Oct-2019.)
Hypotheses
Ref Expression
supfirege.1  |-  ( ph  ->  B  C_  RR )
supfirege.2  |-  ( ph  ->  B  e.  Fin )
supfirege.3  |-  ( ph  ->  C  e.  B )
supfirege.4  |-  ( ph  ->  S  =  sup ( B ,  RR ,  <  ) )
Assertion
Ref Expression
supfirege  |-  ( ph  ->  C  <_  S )

Proof of Theorem supfirege
StepHypRef Expression
1 ltso 9666 . . . 4  |-  <  Or  RR
21a1i 11 . . 3  |-  ( ph  ->  <  Or  RR )
3 supfirege.1 . . 3  |-  ( ph  ->  B  C_  RR )
4 supfirege.2 . . 3  |-  ( ph  ->  B  e.  Fin )
5 supfirege.3 . . 3  |-  ( ph  ->  C  e.  B )
6 supfirege.4 . . 3  |-  ( ph  ->  S  =  sup ( B ,  RR ,  <  ) )
72, 3, 4, 5, 6supgtoreq 7929 . 2  |-  ( ph  ->  ( C  <  S  \/  C  =  S
) )
83, 5sseldd 3505 . . 3  |-  ( ph  ->  C  e.  RR )
9 ne0i 3791 . . . . . . 7  |-  ( C  e.  B  ->  B  =/=  (/) )
105, 9syl 16 . . . . . 6  |-  ( ph  ->  B  =/=  (/) )
11 fisupcl 7928 . . . . . 6  |-  ( (  <  Or  RR  /\  ( B  e.  Fin  /\  B  =/=  (/)  /\  B  C_  RR ) )  ->  sup ( B ,  RR ,  <  )  e.  B
)
122, 4, 10, 3, 11syl13anc 1230 . . . . 5  |-  ( ph  ->  sup ( B ,  RR ,  <  )  e.  B )
136, 12eqeltrd 2555 . . . 4  |-  ( ph  ->  S  e.  B )
143, 13sseldd 3505 . . 3  |-  ( ph  ->  S  e.  RR )
158, 14leloed 9728 . 2  |-  ( ph  ->  ( C  <_  S  <->  ( C  <  S  \/  C  =  S )
) )
167, 15mpbird 232 1  |-  ( ph  ->  C  <_  S )
Colors of variables: wff setvar class
Syntax hints:    -> wi 4    \/ wo 368    = wceq 1379    e. wcel 1767    =/= wne 2662    C_ wss 3476   (/)c0 3785   class class class wbr 4447    Or wor 4799   Fincfn 7517   supcsup 7901   RRcr 9492    < clt 9629    <_ cle 9630
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1601  ax-4 1612  ax-5 1680  ax-6 1719  ax-7 1739  ax-8 1769  ax-9 1771  ax-10 1786  ax-11 1791  ax-12 1803  ax-13 1968  ax-ext 2445  ax-sep 4568  ax-nul 4576  ax-pow 4625  ax-pr 4686  ax-un 6577  ax-resscn 9550  ax-pre-lttri 9567  ax-pre-lttrn 9568
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 974  df-3an 975  df-tru 1382  df-ex 1597  df-nf 1600  df-sb 1712  df-eu 2279  df-mo 2280  df-clab 2453  df-cleq 2459  df-clel 2462  df-nfc 2617  df-ne 2664  df-nel 2665  df-ral 2819  df-rex 2820  df-reu 2821  df-rmo 2822  df-rab 2823  df-v 3115  df-sbc 3332  df-csb 3436  df-dif 3479  df-un 3481  df-in 3483  df-ss 3490  df-pss 3492  df-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-tp 4032  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-tr 4541  df-eprel 4791  df-id 4795  df-po 4800  df-so 4801  df-fr 4838  df-we 4840  df-ord 4881  df-on 4882  df-lim 4883  df-suc 4884  df-xp 5005  df-rel 5006  df-cnv 5007  df-co 5008  df-dm 5009  df-rn 5010  df-res 5011  df-ima 5012  df-iota 5551  df-fun 5590  df-fn 5591  df-f 5592  df-f1 5593  df-fo 5594  df-f1o 5595  df-fv 5596  df-riota 6246  df-om 6686  df-1o 7131  df-er 7312  df-en 7518  df-dom 7519  df-sdom 7520  df-fin 7521  df-sup 7902  df-pnf 9631  df-mnf 9632  df-xr 9633  df-ltxr 9634  df-le 9635
This theorem is referenced by:  fsuppmapnn0fiub  12066
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