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Theorem supxrnemnf 26007
Description: The supremum of a nonempty set of extended reals which does not contain minus infinity is not minus infinity. (Contributed by Thierry Arnoux, 21-Mar-2017.)
Assertion
Ref Expression
supxrnemnf  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  =/= -oo )

Proof of Theorem supxrnemnf
Dummy variable  x is distinct from all other variables.
StepHypRef Expression
1 mnfxr 11086 . . 3  |- -oo  e.  RR*
21a1i 11 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  e.  RR* )
3 supxrcl 11269 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
433ad2ant1 1009 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 simp1 988 . . . 4  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  A  C_ 
RR* )
65, 1jctir 538 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  ( A  C_  RR*  /\ -oo  e.  RR* ) )
7 simpl 457 . . . . . . . 8  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A )  ->  A  C_  RR* )
87sselda 3351 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  RR* )
9 simpr 461 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  A )
10 simplr 754 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -. -oo  e.  A
)
11 nelneq 2536 . . . . . . . 8  |-  ( ( x  e.  A  /\  -. -oo  e.  A )  ->  -.  x  = -oo )
129, 10, 11syl2anc 661 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -.  x  = -oo )
13 ngtmnft 11131 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x  = -oo  <->  -. -oo  <  x ) )
1413biimprd 223 . . . . . . . 8  |-  ( x  e.  RR*  ->  ( -. -oo  <  x  ->  x  = -oo ) )
1514con1d 124 . . . . . . 7  |-  ( x  e.  RR*  ->  ( -.  x  = -oo  -> -oo 
<  x ) )
168, 12, 15sylc 60 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  -> -oo  <  x )
1716reximdva0 3643 . . . . 5  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
18173impa 1182 . . . 4  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
19183com23 1193 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  E. x  e.  A -oo  <  x
)
20 supxrlub 11280 . . . 4  |-  ( ( A  C_  RR*  /\ -oo  e.  RR* )  ->  ( -oo  <  sup ( A ,  RR* ,  <  )  <->  E. x  e.  A -oo  <  x
) )
2120biimprd 223 . . 3  |-  ( ( A  C_  RR*  /\ -oo  e.  RR* )  ->  ( E. x  e.  A -oo  <  x  -> -oo  <  sup ( A ,  RR* ,  <  ) ) )
226, 19, 21sylc 60 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  <  sup ( A ,  RR* ,  <  ) )
23 xrltne 11129 . 2  |-  ( ( -oo  e.  RR*  /\  sup ( A ,  RR* ,  <  )  e.  RR*  /\ -oo  <  sup ( A ,  RR* ,  <  ) )  ->  sup ( A ,  RR* ,  <  )  =/= -oo )
242, 4, 22, 23syl3anc 1218 1  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  =/= -oo )
Colors of variables: wff setvar class
Syntax hints:   -. wn 3    -> wi 4    /\ wa 369    /\ w3a 965    = wceq 1369    e. wcel 1756    =/= wne 2601   E.wrex 2711    C_ wss 3323   (/)c0 3632   class class class wbr 4287   supcsup 7682   -oocmnf 9408   RR*cxr 9409    < clt 9410
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1591  ax-4 1602  ax-5 1670  ax-6 1708  ax-7 1728  ax-8 1758  ax-9 1760  ax-10 1775  ax-11 1780  ax-12 1792  ax-13 1943  ax-ext 2419  ax-sep 4408  ax-nul 4416  ax-pow 4465  ax-pr 4526  ax-un 6367  ax-cnex 9330  ax-resscn 9331  ax-1cn 9332  ax-icn 9333  ax-addcl 9334  ax-addrcl 9335  ax-mulcl 9336  ax-mulrcl 9337  ax-mulcom 9338  ax-addass 9339  ax-mulass 9340  ax-distr 9341  ax-i2m1 9342  ax-1ne0 9343  ax-1rid 9344  ax-rnegex 9345  ax-rrecex 9346  ax-cnre 9347  ax-pre-lttri 9348  ax-pre-lttrn 9349  ax-pre-ltadd 9350  ax-pre-mulgt0 9351  ax-pre-sup 9352
This theorem depends on definitions:  df-bi 185  df-or 370  df-an 371  df-3or 966  df-3an 967  df-tru 1372  df-ex 1587  df-nf 1590  df-sb 1701  df-eu 2256  df-mo 2257  df-clab 2425  df-cleq 2431  df-clel 2434  df-nfc 2563  df-ne 2603  df-nel 2604  df-ral 2715  df-rex 2716  df-reu 2717  df-rmo 2718  df-rab 2719  df-v 2969  df-sbc 3182  df-csb 3284  df-dif 3326  df-un 3328  df-in 3330  df-ss 3337  df-nul 3633  df-if 3787  df-pw 3857  df-sn 3873  df-pr 3875  df-op 3879  df-uni 4087  df-br 4288  df-opab 4346  df-mpt 4347  df-id 4631  df-po 4636  df-so 4637  df-xp 4841  df-rel 4842  df-cnv 4843  df-co 4844  df-dm 4845  df-rn 4846  df-res 4847  df-ima 4848  df-iota 5376  df-fun 5415  df-fn 5416  df-f 5417  df-f1 5418  df-fo 5419  df-f1o 5420  df-fv 5421  df-riota 6047  df-ov 6089  df-oprab 6090  df-mpt2 6091  df-er 7093  df-en 7303  df-dom 7304  df-sdom 7305  df-sup 7683  df-pnf 9412  df-mnf 9413  df-xr 9414  df-ltxr 9415  df-le 9416  df-sub 9589  df-neg 9590
This theorem is referenced by: (None)
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