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Theorem supxrnemnf 28216
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 11403 . . 3  |- -oo  e.  RR*
21a1i 11 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  e.  RR* )
3 supxrcl 11589 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
433ad2ant1 1026 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 simp1 1005 . . . 4  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  A  C_ 
RR* )
65, 1jctir 540 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  ( A  C_  RR*  /\ -oo  e.  RR* ) )
7 simpl 458 . . . . . . . 8  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A )  ->  A  C_  RR* )
87sselda 3461 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  RR* )
9 simpr 462 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  A )
10 simplr 760 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -. -oo  e.  A
)
11 nelneq 2537 . . . . . . . 8  |-  ( ( x  e.  A  /\  -. -oo  e.  A )  ->  -.  x  = -oo )
129, 10, 11syl2anc 665 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -.  x  = -oo )
13 ngtmnft 11451 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x  = -oo  <->  -. -oo  <  x ) )
1413biimprd 226 . . . . . . . 8  |-  ( x  e.  RR*  ->  ( -. -oo  <  x  ->  x  = -oo ) )
1514con1d 127 . . . . . . 7  |-  ( x  e.  RR*  ->  ( -.  x  = -oo  -> -oo 
<  x ) )
168, 12, 15sylc 62 . . . . . 6  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  -> -oo  <  x )
1716reximdva0 3770 . . . . 5  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
18173impa 1200 . . . 4  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
19183com23 1211 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  E. x  e.  A -oo  <  x
)
20 supxrlub 11600 . . . 4  |-  ( ( A  C_  RR*  /\ -oo  e.  RR* )  ->  ( -oo  <  sup ( A ,  RR* ,  <  )  <->  E. x  e.  A -oo  <  x
) )
2120biimprd 226 . . 3  |-  ( ( A  C_  RR*  /\ -oo  e.  RR* )  ->  ( E. x  e.  A -oo  <  x  -> -oo  <  sup ( A ,  RR* ,  <  ) ) )
226, 19, 21sylc 62 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  <  sup ( A ,  RR* ,  <  ) )
23 xrltne 11449 . 2  |-  ( ( -oo  e.  RR*  /\  sup ( A ,  RR* ,  <  )  e.  RR*  /\ -oo  <  sup ( A ,  RR* ,  <  ) )  ->  sup ( A ,  RR* ,  <  )  =/= -oo )
242, 4, 22, 23syl3anc 1264 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 370    /\ w3a 982    = wceq 1437    e. wcel 1867    =/= wne 2616   E.wrex 2774    C_ wss 3433   (/)c0 3758   class class class wbr 4417   supcsup 7951   -oocmnf 9662   RR*cxr 9663    < clt 9664
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1665  ax-4 1678  ax-5 1748  ax-6 1794  ax-7 1838  ax-8 1869  ax-9 1871  ax-10 1886  ax-11 1891  ax-12 1904  ax-13 2052  ax-ext 2398  ax-sep 4539  ax-nul 4547  ax-pow 4594  ax-pr 4652  ax-un 6588  ax-cnex 9584  ax-resscn 9585  ax-1cn 9586  ax-icn 9587  ax-addcl 9588  ax-addrcl 9589  ax-mulcl 9590  ax-mulrcl 9591  ax-mulcom 9592  ax-addass 9593  ax-mulass 9594  ax-distr 9595  ax-i2m1 9596  ax-1ne0 9597  ax-1rid 9598  ax-rnegex 9599  ax-rrecex 9600  ax-cnre 9601  ax-pre-lttri 9602  ax-pre-lttrn 9603  ax-pre-ltadd 9604  ax-pre-mulgt0 9605  ax-pre-sup 9606
This theorem depends on definitions:  df-bi 188  df-or 371  df-an 372  df-3or 983  df-3an 984  df-tru 1440  df-ex 1660  df-nf 1664  df-sb 1787  df-eu 2267  df-mo 2268  df-clab 2406  df-cleq 2412  df-clel 2415  df-nfc 2570  df-ne 2618  df-nel 2619  df-ral 2778  df-rex 2779  df-reu 2780  df-rmo 2781  df-rab 2782  df-v 3080  df-sbc 3297  df-csb 3393  df-dif 3436  df-un 3438  df-in 3440  df-ss 3447  df-nul 3759  df-if 3907  df-pw 3978  df-sn 3994  df-pr 3996  df-op 4000  df-uni 4214  df-br 4418  df-opab 4476  df-mpt 4477  df-id 4760  df-po 4766  df-so 4767  df-xp 4851  df-rel 4852  df-cnv 4853  df-co 4854  df-dm 4855  df-rn 4856  df-res 4857  df-ima 4858  df-iota 5556  df-fun 5594  df-fn 5595  df-f 5596  df-f1 5597  df-fo 5598  df-f1o 5599  df-fv 5600  df-riota 6258  df-ov 6299  df-oprab 6300  df-mpt2 6301  df-er 7362  df-en 7569  df-dom 7570  df-sdom 7571  df-sup 7953  df-pnf 9666  df-mnf 9667  df-xr 9668  df-ltxr 9669  df-le 9670  df-sub 9851  df-neg 9852
This theorem is referenced by: (None)
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