Theorem npnflt 9751

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

Assertion

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

Proof of Theorem npnflt

Step	Hyp	Ref	Expression
1		nltpnft 9750	. . . 4 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )
2	1	biimpd 143	. . 3 |- ( A e. RR* -> ( A = +oo -> -. A < +oo ) )
3	2	necon2ad 2393	. 2 |- ( A e. RR* -> ( A < +oo -> A =/= +oo ) )
4		ltpnf 9716	. . . . 5 |- ( A e. RR -> A < +oo )
5	4	adantl 275	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A e. RR ) -> A < +oo )
6		simpr 109	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A = +oo )
7		simplr 520	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A =/= +oo )
8	6, 7	pm2.21ddne 2419	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A < +oo )
9		mnfltpnf 9721	. . . . . 6 |- -oo < +oo
10		breq1 3985	. . . . . 6 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
11	9, 10	mpbiri 167	. . . . 5 |- ( A = -oo -> A < +oo )
12	11	adantl 275	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = -oo ) -> A < +oo )
13		elxr 9712	. . . . . 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, 8, 12, 15	mpjao3dan 1297	. . 3 |- ( ( A e. RR* /\ A =/= +oo ) -> A < +oo )
17	16	ex 114	. 2 |- ( A e. RR* -> ( A =/= +oo -> A < +oo ) )
18	3, 17	impbid 128	1 |- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 103   <-> wb 104   \/ w3o 967   = wceq 1343   e. wcel 2136   =/= wne 2336   class class class wbr 3982   RRcr 7752   +oocpnf 7930   -oocmnf 7931   RR*cxr 7932   < clt 7933

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 604  ax-in2 605  ax-io 699  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-10 1493  ax-11 1494  ax-i12 1495  ax-bndl 1497  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-13 2138  ax-14 2139  ax-ext 2147  ax-sep 4100  ax-pow 4153  ax-pr 4187  ax-un 4411  ax-setind 4514  ax-cnex 7844  ax-resscn 7845  ax-pre-ltirr 7865

This theorem depends on definitions:  df-bi 116  df-3or 969  df-3an 970  df-tru 1346  df-fal 1349  df-nf 1449  df-sb 1751  df-eu 2017  df-mo 2018  df-clab 2152  df-cleq 2158  df-clel 2161  df-nfc 2297  df-ne 2337  df-nel 2432  df-ral 2449  df-rex 2450  df-rab 2453  df-v 2728  df-dif 3118  df-un 3120  df-in 3122  df-ss 3129  df-pw 3561  df-sn 3582  df-pr 3583  df-op 3585  df-uni 3790  df-br 3983  df-opab 4044  df-xp 4610  df-pnf 7935  df-mnf 7936  df-xr 7937  df-ltxr 7938

This theorem is referenced by:  xlt2add  9816  xrmaxadd  11202