Theorem npnflt 9972

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 9971	. . . 4 |- ( A e. RR* -> ( A = +oo <-> -. A < +oo ) )
2	1	biimpd 144	. . 3 |- ( A e. RR* -> ( A = +oo -> -. A < +oo ) )
3	2	necon2ad 2435	. 2 |- ( A e. RR* -> ( A < +oo -> A =/= +oo ) )
4		ltpnf 9937	. . . . 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 528	. . . . 5 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A =/= +oo )
8	6, 7	pm2.21ddne 2461	. . . 4 |- ( ( ( A e. RR* /\ A =/= +oo ) /\ A = +oo ) -> A < +oo )
9		mnfltpnf 9942	. . . . . 6 |- -oo < +oo
10		breq1 4062	. . . . . 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 9933	. . . . . 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 1320	. . 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 980   = wceq 1373   e. wcel 2178   =/= wne 2378   class class class wbr 4059   RRcr 7959   +oocpnf 8139   -oocmnf 8140   RR*cxr 8141   < clt 8142

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-13 2180  ax-14 2181  ax-ext 2189  ax-sep 4178  ax-pow 4234  ax-pr 4269  ax-un 4498  ax-setind 4603  ax-cnex 8051  ax-resscn 8052  ax-pre-ltirr 8072

This theorem depends on definitions:  df-bi 117  df-3or 982  df-3an 983  df-tru 1376  df-fal 1379  df-nf 1485  df-sb 1787  df-eu 2058  df-mo 2059  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-ne 2379  df-nel 2474  df-ral 2491  df-rex 2492  df-rab 2495  df-v 2778  df-dif 3176  df-un 3178  df-in 3180  df-ss 3187  df-pw 3628  df-sn 3649  df-pr 3650  df-op 3652  df-uni 3865  df-br 4060  df-opab 4122  df-xp 4699  df-pnf 8144  df-mnf 8145  df-xr 8146  df-ltxr 8147

This theorem is referenced by:  xlt2add  10037  xrmaxadd  11687