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Theorem supxrnemnf 28366
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 11421 . . 3  |- -oo  e.  RR*
21a1i 11 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  -> -oo  e.  RR* )
3 supxrcl 11607 . . 3  |-  ( A 
C_  RR*  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
433ad2ant1 1030 . 2  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  sup ( A ,  RR* ,  <  )  e.  RR* )
5 simp1 1009 . . . 4  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  A  C_ 
RR* )
65, 1jctir 541 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  ( A  C_  RR*  /\ -oo  e.  RR* ) )
7 simpl 459 . . . . . . . 8  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A )  ->  A  C_  RR* )
87sselda 3434 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  RR* )
9 simpr 463 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  x  e.  A )
10 simplr 763 . . . . . . . 8  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -. -oo  e.  A
)
11 nelneq 2555 . . . . . . . 8  |-  ( ( x  e.  A  /\  -. -oo  e.  A )  ->  -.  x  = -oo )
129, 10, 11syl2anc 667 . . . . . . 7  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  x  e.  A )  ->  -.  x  = -oo )
13 ngtmnft 11469 . . . . . . . . 9  |-  ( x  e.  RR*  ->  ( x  = -oo  <->  -. -oo  <  x ) )
1413biimprd 227 . . . . . . . 8  |-  ( x  e.  RR*  ->  ( -. -oo  <  x  ->  x  = -oo ) )
1514con1d 128 . . . . . . 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 3745 . . . . 5  |-  ( ( ( A  C_  RR*  /\  -. -oo  e.  A )  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
18173impa 1204 . . . 4  |-  ( ( A  C_  RR*  /\  -. -oo  e.  A  /\  A  =/=  (/) )  ->  E. x  e.  A -oo  <  x
)
19183com23 1215 . . 3  |-  ( ( A  C_  RR*  /\  A  =/=  (/)  /\  -. -oo  e.  A )  ->  E. x  e.  A -oo  <  x
)
20 supxrlub 11618 . . . 4  |-  ( ( A  C_  RR*  /\ -oo  e.  RR* )  ->  ( -oo  <  sup ( A ,  RR* ,  <  )  <->  E. x  e.  A -oo  <  x
) )
2120biimprd 227 . . 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 11467 . 2  |-  ( ( -oo  e.  RR*  /\  sup ( A ,  RR* ,  <  )  e.  RR*  /\ -oo  <  sup ( A ,  RR* ,  <  ) )  ->  sup ( A ,  RR* ,  <  )  =/= -oo )
242, 4, 22, 23syl3anc 1269 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 371    /\ w3a 986    = wceq 1446    e. wcel 1889    =/= wne 2624   E.wrex 2740    C_ wss 3406   (/)c0 3733   class class class wbr 4405   supcsup 7959   -oocmnf 9678   RR*cxr 9679    < clt 9680
This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1671  ax-4 1684  ax-5 1760  ax-6 1807  ax-7 1853  ax-8 1891  ax-9 1898  ax-10 1917  ax-11 1922  ax-12 1935  ax-13 2093  ax-ext 2433  ax-sep 4528  ax-nul 4537  ax-pow 4584  ax-pr 4642  ax-un 6588  ax-cnex 9600  ax-resscn 9601  ax-1cn 9602  ax-icn 9603  ax-addcl 9604  ax-addrcl 9605  ax-mulcl 9606  ax-mulrcl 9607  ax-mulcom 9608  ax-addass 9609  ax-mulass 9610  ax-distr 9611  ax-i2m1 9612  ax-1ne0 9613  ax-1rid 9614  ax-rnegex 9615  ax-rrecex 9616  ax-cnre 9617  ax-pre-lttri 9618  ax-pre-lttrn 9619  ax-pre-ltadd 9620  ax-pre-mulgt0 9621  ax-pre-sup 9622
This theorem depends on definitions:  df-bi 189  df-or 372  df-an 373  df-3or 987  df-3an 988  df-tru 1449  df-ex 1666  df-nf 1670  df-sb 1800  df-eu 2305  df-mo 2306  df-clab 2440  df-cleq 2446  df-clel 2449  df-nfc 2583  df-ne 2626  df-nel 2627  df-ral 2744  df-rex 2745  df-reu 2746  df-rmo 2747  df-rab 2748  df-v 3049  df-sbc 3270  df-csb 3366  df-dif 3409  df-un 3411  df-in 3413  df-ss 3420  df-nul 3734  df-if 3884  df-pw 3955  df-sn 3971  df-pr 3973  df-op 3977  df-uni 4202  df-br 4406  df-opab 4465  df-mpt 4466  df-id 4752  df-po 4758  df-so 4759  df-xp 4843  df-rel 4844  df-cnv 4845  df-co 4846  df-dm 4847  df-rn 4848  df-res 4849  df-ima 4850  df-iota 5549  df-fun 5587  df-fn 5588  df-f 5589  df-f1 5590  df-fo 5591  df-f1o 5592  df-fv 5593  df-riota 6257  df-ov 6298  df-oprab 6299  df-mpt2 6300  df-er 7368  df-en 7575  df-dom 7576  df-sdom 7577  df-sup 7961  df-pnf 9682  df-mnf 9683  df-xr 9684  df-ltxr 9685  df-le 9686  df-sub 9867  df-neg 9868
This theorem is referenced by: (None)
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