Theorem nmnfgt 9601

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 9600	. . . 4 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )
2	1	biimpd 143	. . 3 |- ( A e. RR* -> ( A = -oo -> -. -oo < A ) )
3	2	necon2ad 2365	. 2 |- ( A e. RR* -> ( -oo < A -> A =/= -oo ) )
4		mnflt 9569	. . . . 5 |- ( A e. RR -> -oo < A )
5	4	adantl 275	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A e. RR ) -> -oo < A )
6		mnfltpnf 9571	. . . . . 6 |- -oo < +oo
7		breq2 3933	. . . . . 6 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
8	6, 7	mpbiri 167	. . . . 5 |- ( A = +oo -> -oo < A )
9	8	adantl 275	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = +oo ) -> -oo < A )
10		simpr 109	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A = -oo )
11		simplr 519	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A =/= -oo )
12	10, 11	pm2.21ddne 2391	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> -oo < A )
13		elxr 9563	. . . . . 6 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
14	13	biimpi 119	. . . . 5 |- ( A e. RR* -> ( A e. RR \/ A = +oo \/ A = -oo ) )
15	14	adantr 274	. . . 4 |- ( ( A e. RR* /\ A =/= -oo ) -> ( A e. RR \/ A = +oo \/ A = -oo ) )
16	5, 9, 12, 15	mpjao3dan 1285	. . 3 |- ( ( A e. RR* /\ A =/= -oo ) -> -oo < A )
17	16	ex 114	. 2 |- ( A e. RR* -> ( A =/= -oo -> -oo < A ) )
18	3, 17	impbid 128	1 |- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 961   = wceq 1331   e. wcel 1480   =/= wne 2308   class class class wbr 3929   RRcr 7619   +oocpnf 7797   -oocmnf 7798   RR*cxr 7799   < clt 7800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 603  ax-in2 604  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-13 1491  ax-14 1492  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121  ax-sep 4046  ax-pow 4098  ax-pr 4131  ax-un 4355  ax-setind 4452  ax-cnex 7711  ax-resscn 7712  ax-pre-ltirr 7732

This theorem depends on definitions:  df-bi 116  df-3or 963  df-3an 964  df-tru 1334  df-fal 1337  df-nf 1437  df-sb 1736  df-eu 2002  df-mo 2003  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-ne 2309  df-nel 2404  df-ral 2421  df-rex 2422  df-rab 2425  df-v 2688  df-dif 3073  df-un 3075  df-in 3077  df-ss 3084  df-pw 3512  df-sn 3533  df-pr 3534  df-op 3536  df-uni 3737  df-br 3930  df-opab 3990  df-xp 4545  df-pnf 7802  df-mnf 7803  df-xr 7804  df-ltxr 7805

This theorem is referenced by:  xlt2add  9663  xrmaxadd  11030  xblpnfps  12567  xblpnf  12568