Theorem nmnfgt 9884

Description: An extended real is greater than minus infinite iff they are not equal. (Contributed by Jim Kingdon, 17-Apr-2023.)

Assertion

Ref	Expression
nmnfgt	|- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )

Proof of Theorem nmnfgt

Step	Hyp	Ref	Expression
1		ngtmnft 9883	. . . 4 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )
2	1	biimpd 144	. . 3 |- ( A e. RR* -> ( A = -oo -> -. -oo < A ) )
3	2	necon2ad 2421	. 2 |- ( A e. RR* -> ( -oo < A -> A =/= -oo ) )
4		mnflt 9849	. . . . 5 |- ( A e. RR -> -oo < A )
5	4	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A e. RR ) -> -oo < A )
6		mnfltpnf 9851	. . . . . 6 |- -oo < +oo
7		breq2 4033	. . . . . 6 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
8	6, 7	mpbiri 168	. . . . 5 |- ( A = +oo -> -oo < A )
9	8	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = +oo ) -> -oo < A )
10		simpr 110	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A = -oo )
11		simplr 528	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A =/= -oo )
12	10, 11	pm2.21ddne 2447	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> -oo < A )
13		elxr 9842	. . . . . 6 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
14	13	biimpi 120	. . . . 5 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
15	14	adantr 276	. . . 4 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
16	5, 9, 12, 15	mpjao3dan 1318	. . 3 |- ( ( A e. RR* /\ A =/= -oo ) -> -oo < A )
17	16	ex 115	. 2 |- ( A e. RR* -> ( A =/= -oo -> -oo < A ) )
18	3, 17	impbid 129	1 |- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 979   = wceq 1364   e. wcel 2164   =/= wne 2364   class class class wbr 4029   RRcr 7871   +oocpnf 8051   -oocmnf 8052   RR*cxr 8053   < clt 8054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-in1 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238  ax-un 4464  ax-setind 4569  ax-cnex 7963  ax-resscn 7964  ax-pre-ltirr 7984

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3155  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-uni 3836  df-br 4030  df-opab 4091  df-xp 4665  df-pnf 8056  df-mnf 8057  df-xr 8058  df-ltxr 8059

This theorem is referenced by:  xlt2add  9946  xrmaxadd  11404  xblpnfps  14566  xblpnf  14567