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Theorem supxrnemnf 28429
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 11437 . . 3  |- -oo  e.  RR*
21a1i 11 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  e.  RR* )
3 supxrcl 11625 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
433ad2ant1 1051 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 simp1 1030 . . . 4  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  A  C_ 
RR* )
65, 1jctir 547 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  ( A  C_  RR*  /\ -oo  e.  RR* ) )
7 simpl 464 . . . . . . . 8  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A )  ->  A  C_  RR* )
87sselda 3418 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  RR* )
9 simpr 468 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  A )
10 simplr 770 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -. -oo  e.  A
)
11 nelneq 2573 . . . . . . . 8  |-  ( ( x  e.  A  /\  -. -oo  e.  A )  ->  -.  x  = -oo )
129, 10, 11syl2anc 673 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -.  x  = -oo )
13 ngtmnft 11485 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x  = -oo  <->  -. -oo  <  x ) )
1413biimprd 231 . . . . . . . 8  |-  ( x  e.  RR*  ->  ( -. -oo  <  x  ->  x  = -oo ) )
1514con1d 129 . . . . . . 7  |-  ( x  e.  RR*  ->  ( -.  x  = -oo  -> -oo 
<  x ) )
168, 12, 15sylc 61 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  -> -oo  <  x )
1716reximdva0 3734 . . . . 5  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
18173impa 1226 . . . 4  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
19183com23 1237 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  E. x  e.  A -oo  <  x
)
20 supxrlub 11636 . . . 4  |-  ( ( A  C_  RR*  /\ -oo  e.  RR* )  ->  ( -oo  <  sup ( A ,  RR* ,  <  )  <->  E. x  e.  A -oo  <  x
) )
2120biimprd 231 . . 3  |-  ( ( A  C_  RR*  /\ -oo  e.  RR* )  ->  ( E. x  e.  A -oo  <  x  -> -oo  <  sup ( A ,  RR* ,  <  ) ) )
226, 19, 21sylc 61 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  <  sup ( A ,  RR* ,  <  ) )
23 xrltne 11483 . 2  |-  ( ( -oo  e.  RR*  /\  sup ( A ,  RR* ,  <  )  e.  RR*  /\ -oo  <  sup ( A ,  RR* ,  <  ) )  ->  sup ( A ,  RR* ,  <  )  =/= -oo )
242, 4, 22, 23syl3anc 1292 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 376    /\ w3a 1007    = wceq 1452    e. wcel 1904    =/= wne 2641   E.wrex 2757    C_ wss 3390   (/)c0 3722   class class class wbr 4395   supcsup 7972   -oocmnf 9691   RR*cxr 9692    < clt 9693
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1677  ax-4 1690  ax-5 1766  ax-6 1813  ax-7 1859  ax-8 1906  ax-9 1913  ax-10 1932  ax-11 1937  ax-12 1950  ax-13 2104  ax-ext 2451  ax-sep 4518  ax-nul 4527  ax-pow 4579  ax-pr 4639  ax-un 6602  ax-cnex 9613  ax-resscn 9614  ax-1cn 9615  ax-icn 9616  ax-addcl 9617  ax-addrcl 9618  ax-mulcl 9619  ax-mulrcl 9620  ax-mulcom 9621  ax-addass 9622  ax-mulass 9623  ax-distr 9624  ax-i2m1 9625  ax-1ne0 9626  ax-1rid 9627  ax-rnegex 9628  ax-rrecex 9629  ax-cnre 9630  ax-pre-lttri 9631  ax-pre-lttrn 9632  ax-pre-ltadd 9633  ax-pre-mulgt0 9634  ax-pre-sup 9635
This theorem depends on definitions:  df-bi 190  df-or 377  df-an 378  df-3or 1008  df-3an 1009  df-tru 1455  df-ex 1672  df-nf 1676  df-sb 1806  df-eu 2323  df-mo 2324  df-clab 2458  df-cleq 2464  df-clel 2467  df-nfc 2601  df-ne 2643  df-nel 2644  df-ral 2761  df-rex 2762  df-reu 2763  df-rmo 2764  df-rab 2765  df-v 3033  df-sbc 3256  df-csb 3350  df-dif 3393  df-un 3395  df-in 3397  df-ss 3404  df-nul 3723  df-if 3873  df-pw 3944  df-sn 3960  df-pr 3962  df-op 3966  df-uni 4191  df-br 4396  df-opab 4455  df-mpt 4456  df-id 4754  df-po 4760  df-so 4761  df-xp 4845  df-rel 4846  df-cnv 4847  df-co 4848  df-dm 4849  df-rn 4850  df-res 4851  df-ima 4852  df-iota 5553  df-fun 5591  df-fn 5592  df-f 5593  df-f1 5594  df-fo 5595  df-f1o 5596  df-fv 5597  df-riota 6270  df-ov 6311  df-oprab 6312  df-mpt2 6313  df-er 7381  df-en 7588  df-dom 7589  df-sdom 7590  df-sup 7974  df-pnf 9695  df-mnf 9696  df-xr 9697  df-ltxr 9698  df-le 9699  df-sub 9882  df-neg 9883
This theorem is referenced by: (None)
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