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Metamath Proof Explorer

Theorem supxrnemnf

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)

Ref Expression
Assertion supxrnemnf 
|- ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> sup ( A , RR* , < ) =/= -oo )

Proof

Step Hyp Ref Expression
1 mnfxr 
 |-  -oo e. RR*
2 1 a1i 
 |-  ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> -oo e. RR* )
3 supxrcl 
 |-  ( A C_ RR* -> sup ( A , RR* , < ) e. RR* )
4 3 3ad2ant1 
 |-  ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> sup ( A , RR* , < ) e. RR* )
5 simp1 
 |-  ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> A C_ RR* )
6 5 1 jctir 
 |-  ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> ( A C_ RR* /\ -oo e. RR* ) )
7 simpl 
 |-  ( ( A C_ RR* /\ -. -oo e. A ) -> A C_ RR* )
8 7 sselda 
 |-  ( ( ( A C_ RR* /\ -. -oo e. A ) /\ x e. A ) -> x e. RR* )
9 simpr 
 |-  ( ( ( A C_ RR* /\ -. -oo e. A ) /\ x e. A ) -> x e. A )
10 simplr 
 |-  ( ( ( A C_ RR* /\ -. -oo e. A ) /\ x e. A ) -> -. -oo e. A )
11 nelneq 
 |-  ( ( x e. A /\ -. -oo e. A ) -> -. x = -oo )
12 9 10 11 syl2anc 
 |-  ( ( ( A C_ RR* /\ -. -oo e. A ) /\ x e. A ) -> -. x = -oo )
13 ngtmnft 
 |-  ( x e. RR* -> ( x = -oo <-> -. -oo < x ) )
14 13 biimprd 
 |-  ( x e. RR* -> ( -. -oo < x -> x = -oo ) )
15 14 con1d 
 |-  ( x e. RR* -> ( -. x = -oo -> -oo < x ) )
16 8 12 15 sylc 
 |-  ( ( ( A C_ RR* /\ -. -oo e. A ) /\ x e. A ) -> -oo < x )
17 16 reximdva0 
 |-  ( ( ( A C_ RR* /\ -. -oo e. A ) /\ A =/= (/) ) -> E. x e. A -oo < x )
18 17 3impa 
 |-  ( ( A C_ RR* /\ -. -oo e. A /\ A =/= (/) ) -> E. x e. A -oo < x )
19 18 3com23 
 |-  ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> E. x e. A -oo < x )
20 supxrlub 
 |-  ( ( A C_ RR* /\ -oo e. RR* ) -> ( -oo < sup ( A , RR* , < ) <-> E. x e. A -oo < x ) )
21 20 biimprd 
 |-  ( ( A C_ RR* /\ -oo e. RR* ) -> ( E. x e. A -oo < x -> -oo < sup ( A , RR* , < ) ) )
22 6 19 21 sylc 
 |-  ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> -oo < sup ( A , RR* , < ) )
23 xrltne 
 |-  ( ( -oo e. RR* /\ sup ( A , RR* , < ) e. RR* /\ -oo < sup ( A , RR* , < ) ) -> sup ( A , RR* , < ) =/= -oo )
24 2 4 22 23 syl3anc 
 |-  ( ( A C_ RR* /\ A =/= (/) /\ -. -oo e. A ) -> sup ( A , RR* , < ) =/= -oo )