Theorem nmnfgt 9820

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 9819	. . . 4 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )
2	1	biimpd 144	. . 3 |- ( A e. RR* -> ( A = -oo -> -. -oo < A ) )
3	2	necon2ad 2404	. 2 |- ( A e. RR* -> ( -oo < A -> A =/= -oo ) )
4		mnflt 9785	. . . . 5 |- ( A e. RR -> -oo < A )
5	4	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A e. RR ) -> -oo < A )
6		mnfltpnf 9787	. . . . . 6 |- -oo < +oo
7		breq2 4009	. . . . . 6 |- ( A = +oo -> ( -oo < A <-> -oo < +oo ) )
8	6, 7	mpbiri 168	. . . . 5 |- ( A = +oo -> -oo < A )
9	8	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = +oo ) -> -oo < A )
10		simpr 110	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A = -oo )
11		simplr 528	. . . . 5 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> A =/= -oo )
12	10, 11	pm2.21ddne 2430	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> -oo < A )
13		elxr 9778	. . . . . 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, 9, 12, 15	mpjao3dan 1307	. . 3 |- ( ( A e. RR* /\ A =/= -oo ) -> -oo < A )
17	16	ex 115	. 2 |- ( A e. RR* -> ( A =/= -oo -> -oo < A ) )
18	3, 17	impbid 129	1 |- ( A e. RR* -> ( -oo < A <-> A =/= -oo ) )

Syntax hints:   -. wn 3   -> wi 4   /\ wa 104   <-> wb 105   \/ w3o 977   = wceq 1353   e. wcel 2148   =/= wne 2347   class class class wbr 4005   RRcr 7812   +oocpnf 7991   -oocmnf 7992   RR*cxr 7993   < clt 7994

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 614  ax-in2 615  ax-io 709  ax-5 1447  ax-7 1448  ax-gen 1449  ax-ie1 1493  ax-ie2 1494  ax-8 1504  ax-10 1505  ax-11 1506  ax-i12 1507  ax-bndl 1509  ax-4 1510  ax-17 1526  ax-i9 1530  ax-ial 1534  ax-i5r 1535  ax-13 2150  ax-14 2151  ax-ext 2159  ax-sep 4123  ax-pow 4176  ax-pr 4211  ax-un 4435  ax-setind 4538  ax-cnex 7904  ax-resscn 7905  ax-pre-ltirr 7925

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1461  df-sb 1763  df-eu 2029  df-mo 2030  df-clab 2164  df-cleq 2170  df-clel 2173  df-nfc 2308  df-ne 2348  df-nel 2443  df-ral 2460  df-rex 2461  df-rab 2464  df-v 2741  df-dif 3133  df-un 3135  df-in 3137  df-ss 3144  df-pw 3579  df-sn 3600  df-pr 3601  df-op 3603  df-uni 3812  df-br 4006  df-opab 4067  df-xp 4634  df-pnf 7996  df-mnf 7997  df-xr 7998  df-ltxr 7999

This theorem is referenced by:  xlt2add  9882  xrmaxadd  11271  xblpnfps  13983  xblpnf  13984