Theorem npnflt 9742

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 9741	. . . 4 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )
2	1	biimpd 143	. . 3 |- ( A e. RR* -> ( A = +oo -> -. A < +oo ) )
3	2	necon2ad 2391	. 2 |- ( A e. RR* -> ( A < +oo -> A =/= +oo ) )
4		ltpnf 9707	. . . . 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 2417	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A < +oo )
9		mnfltpnf 9712	. . . . . 6 |- -oo < +oo
10		breq1 3979	. . . . . 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 9703	. . . . . 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 1296	. . 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 966   = wceq 1342   e. wcel 2135   =/= wne 2334   class class class wbr 3976   RRcr 7743   +oocpnf 7921   -oocmnf 7922   RR*cxr 7923   < clt 7924

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 1434  ax-7 1435  ax-gen 1436  ax-ie1 1480  ax-ie2 1481  ax-8 1491  ax-10 1492  ax-11 1493  ax-i12 1494  ax-bndl 1496  ax-4 1497  ax-17 1513  ax-i9 1517  ax-ial 1521  ax-i5r 1522  ax-13 2137  ax-14 2138  ax-ext 2146  ax-sep 4094  ax-pow 4147  ax-pr 4181  ax-un 4405  ax-setind 4508  ax-cnex 7835  ax-resscn 7836  ax-pre-ltirr 7856

This theorem depends on definitions:  df-bi 116  df-3or 968  df-3an 969  df-tru 1345  df-fal 1348  df-nf 1448  df-sb 1750  df-eu 2016  df-mo 2017  df-clab 2151  df-cleq 2157  df-clel 2160  df-nfc 2295  df-ne 2335  df-nel 2430  df-ral 2447  df-rex 2448  df-rab 2451  df-v 2723  df-dif 3113  df-un 3115  df-in 3117  df-ss 3124  df-pw 3555  df-sn 3576  df-pr 3577  df-op 3579  df-uni 3784  df-br 3977  df-opab 4038  df-xp 4604  df-pnf 7926  df-mnf 7927  df-xr 7928  df-ltxr 7929

This theorem is referenced by:  xlt2add  9807  xrmaxadd  11188