Users' Mathboxes Mathbox for Thierry Arnoux < Previous   Next >
Nearby theorems
Mirrors  >  Home  >  MPE Home  >  Th. List  >   Mathboxes  >  supxrnemnf Structured version   Unicode version

Theorem supxrnemnf 27248
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 11319 . . 3  |- -oo  e.  RR*
21a1i 11 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  e.  RR* )
3 supxrcl 11502 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
433ad2ant1 1017 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 simp1 996 . . . 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 3504 . . . . . . 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 2584 . . . . . . . 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 11364 . . . . . . . . 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 3796 . . . . 5  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
18173impa 1191 . . . 4  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
19183com23 1202 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  E. x  e.  A -oo  <  x
)
20 supxrlub 11513 . . . 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 11362 . 2  |-  ( ( -oo  e.  RR*  /\  sup ( A ,  RR* ,  <  )  e.  RR*  /\ -oo  <  sup ( A ,  RR* ,  <  ) )  ->  sup ( A ,  RR* ,  <  )  =/= -oo )
242, 4, 22, 23syl3anc 1228 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 973    = wceq 1379    e. wcel 1767    =/= wne 2662   E.wrex 2815    C_ wss 3476   (/)c0 3785   class class class wbr 4447   supcsup 7896   -oocmnf 9622   RR*cxr 9623    < clt 9624
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 6574  ax-cnex 9544  ax-resscn 9545  ax-1cn 9546  ax-icn 9547  ax-addcl 9548  ax-addrcl 9549  ax-mulcl 9550  ax-mulrcl 9551  ax-mulcom 9552  ax-addass 9553  ax-mulass 9554  ax-distr 9555  ax-i2m1 9556  ax-1ne0 9557  ax-1rid 9558  ax-rnegex 9559  ax-rrecex 9560  ax-cnre 9561  ax-pre-lttri 9562  ax-pre-lttrn 9563  ax-pre-ltadd 9564  ax-pre-mulgt0 9565  ax-pre-sup 9566
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-nul 3786  df-if 3940  df-pw 4012  df-sn 4028  df-pr 4030  df-op 4034  df-uni 4246  df-br 4448  df-opab 4506  df-mpt 4507  df-id 4795  df-po 4800  df-so 4801  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 5549  df-fun 5588  df-fn 5589  df-f 5590  df-f1 5591  df-fo 5592  df-f1o 5593  df-fv 5594  df-riota 6243  df-ov 6285  df-oprab 6286  df-mpt2 6287  df-er 7308  df-en 7514  df-dom 7515  df-sdom 7516  df-sup 7897  df-pnf 9626  df-mnf 9627  df-xr 9628  df-ltxr 9629  df-le 9630  df-sub 9803  df-neg 9804
This theorem is referenced by: (None)
  Copyright terms: Public domain W3C validator