Theorem nmnfgt 9789

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 9788	. . . 4 |- ( A e. RR* -> ( A = -oo <-> -. -oo < A ) )
2	1	biimpd 144	. . 3 |- ( A e. RR* -> ( A = -oo -> -. -oo < A ) )
3	2	necon2ad 2402	. 2 |- ( A e. RR* -> ( -oo < A -> A =/= -oo ) )
4		mnflt 9754	. . . . 5 |- ( A e. RR -> -oo < A )
5	4	adantl 277	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A e. RR ) -> -oo < A )
6		mnfltpnf 9756	. . . . . 6 |- -oo < +oo
7		breq2 4002	. . . . . 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 2428	. . . 4 |- ( ( ( A e. RR* /\ A =/= -oo ) /\ A = -oo ) -> -oo < A )
13		elxr 9747	. . . . . 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 2146   =/= wne 2345   class class class wbr 3998   RRcr 7785   +oocpnf 7963   -oocmnf 7964   RR*cxr 7965   < clt 7966

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 1445  ax-7 1446  ax-gen 1447  ax-ie1 1491  ax-ie2 1492  ax-8 1502  ax-10 1503  ax-11 1504  ax-i12 1505  ax-bndl 1507  ax-4 1508  ax-17 1524  ax-i9 1528  ax-ial 1532  ax-i5r 1533  ax-13 2148  ax-14 2149  ax-ext 2157  ax-sep 4116  ax-pow 4169  ax-pr 4203  ax-un 4427  ax-setind 4530  ax-cnex 7877  ax-resscn 7878  ax-pre-ltirr 7898

This theorem depends on definitions:  df-bi 117  df-3or 979  df-3an 980  df-tru 1356  df-fal 1359  df-nf 1459  df-sb 1761  df-eu 2027  df-mo 2028  df-clab 2162  df-cleq 2168  df-clel 2171  df-nfc 2306  df-ne 2346  df-nel 2441  df-ral 2458  df-rex 2459  df-rab 2462  df-v 2737  df-dif 3129  df-un 3131  df-in 3133  df-ss 3140  df-pw 3574  df-sn 3595  df-pr 3596  df-op 3598  df-uni 3806  df-br 3999  df-opab 4060  df-xp 4626  df-pnf 7968  df-mnf 7969  df-xr 7970  df-ltxr 7971

This theorem is referenced by:  xlt2add  9851  xrmaxadd  11237  xblpnfps  13469  xblpnf  13470