Theorem npnflt 10167

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 10166	. . . 4 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )
2	1	biimpd 144	. . 3 |- ( A e. RR* -> ( A = +oo -> -. A < +oo ) )
3	2	necon2ad 2471	. 2 |- ( A e. RR* -> ( A < +oo -> A =/= +oo ) )
4		ltpnf 10132	. . . . 5 |- ( A e. RR -> A < +oo )
5	4	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A e. RR ) -> A < +oo )
6		simpr 110	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A = +oo )
7		simplr 529	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A =/= +oo )
8	6, 7	pm2.21ddne 2497	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A < +oo )
9		mnfltpnf 10137	. . . . . 6 |- -oo < +oo
10		breq1 4117	. . . . . 6 |- ( A = -oo -> ( A < +oo <-> -oo < +oo ) )
11	9, 10	mpbiri 168	. . . . 5 |- ( A = -oo -> A < +oo )
12	11	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = -oo ) -> A < +oo )
13		elxr 10128	. . . . . 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, 8, 12, 15	mpjao3dan 1344	. . 3 |- ( ( A e. RR* /\ A =/= +oo ) -> A < +oo )
17	16	ex 115	. 2 |- ( A e. RR* -> ( A =/= +oo -> A < +oo ) )
18	3, 17	impbid 129	1 |- ( A e. RR* -> ( A < +oo <-> A =/= +oo ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 1004   = wceq 1398   e. wcel 2205   =/= wne 2414   class class class wbr 4114   RRcr 8142   +oocpnf 8321   -oocmnf 8322   RR*cxr 8323   < clt 8324

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4233  ax-pow 4292  ax-pr 4327  ax-un 4559  ax-setind 4664  ax-cnex 8234  ax-resscn 8235  ax-pre-ltirr 8255

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3216  df-un 3218  df-in 3220  df-ss 3227  df-pw 3676  df-sn 3700  df-pr 3701  df-op 3703  df-uni 3920  df-br 4115  df-opab 4177  df-xp 4760  df-pnf 8326  df-mnf 8327  df-xr 8328  df-ltxr 8329

This theorem is referenced by:  xlt2add  10232  xrmaxadd  11971