Theorem xrmnfdc 10224

Description: An extended real is or is not minus infinity. (Contributed by Jim Kingdon, 13-Apr-2023.)

Assertion

Ref	Expression
xrmnfdc	|- ( A e. RR* -> DECID A = -oo )

Proof of Theorem xrmnfdc

Step	Hyp	Ref	Expression
1		elxr 10157	. 2 |- ( A e. RR* <-> ( A e. RR \/ A = +oo \/ A = -oo ) )
2		renemnf 8364	. . . . . 6 |- ( A e. RR -> A =/= -oo )
3	2	neneqd 2441	. . . . 5 |- ( A e. RR -> -. A = -oo )
4	3	olcd 746	. . . 4 |- ( A e. RR -> ( A = -oo \/ -. A = -oo ) )
5		df-dc 847	. . . 4 |- (DECID A = -oo <-> ( A = -oo \/ -. A = -oo ) )
6	4, 5	sylibr 134	. . 3 |- ( A e. RR -> DECID A = -oo )
7		pnfnemnf 8370	. . . . . . 7 |- +oo =/= -oo
8	7	neii 2422	. . . . . 6 |- -. +oo = -oo
9		eqeq1 2245	. . . . . 6 |- ( A = +oo -> ( A = -oo <-> +oo = -oo ) )
10	8, 9	mtbiri 686	. . . . 5 |- ( A = +oo -> -. A = -oo )
11	10	olcd 746	. . . 4 |- ( A = +oo -> ( A = -oo \/ -. A = -oo ) )
12	11, 5	sylibr 134	. . 3 |- ( A = +oo -> DECID A = -oo )
13		orc 724	. . . 4 |- ( A = -oo -> ( A = -oo \/ -. A = -oo ) )
14	13, 5	sylibr 134	. . 3 |- ( A = -oo -> DECID A = -oo )
15	6, 12, 14	3jaoi 1344	. 2 |- ( ( A e. RR \/ A = +oo \/ A = -oo ) -> DECID A = -oo )
16	1, 15	sylbi 121	1 |- ( A e. RR* -> DECID A = -oo )

Syntax hints:   -. wn 3   -> wi 4   \/ wo 720  DECID wdc 846   \/ w3o 1008   = wceq 1402   e. wcel 2209   RRcr 8168   +oocpnf 8347   -oocmnf 8348   RR*cxr 8349

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 623  ax-in2 624  ax-io 721  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-10 1558  ax-11 1559  ax-i12 1560  ax-bndl 1562  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-14 2212  ax-ext 2220  ax-sep 4244  ax-pow 4306  ax-un 4573  ax-setind 4679  ax-cnex 8260  ax-resscn 8261

This theorem depends on definitions:  df-bi 117  df-dc 847  df-3or 1010  df-3an 1011  df-tru 1405  df-fal 1408  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-nfc 2381  df-ne 2421  df-nel 2516  df-ral 2533  df-rex 2534  df-rab 2537  df-v 2823  df-dif 3222  df-un 3224  df-in 3226  df-ss 3233  df-pw 3687  df-sn 3711  df-pr 3712  df-uni 3931  df-pnf 8352  df-mnf 8353  df-xr 8354

This theorem is referenced by:  xaddf  10225  xaddval  10226  xaddmnf1  10229  xaddcom  10242  xnegdi  10249  xpncan  10252  xleadd1a  10254  xsubge0  10262  xrmaxiflemcl  11989  xrmaxifle  11990  xrmaxiflemab  11991  xrmaxiflemlub  11992  xrmaxiflemcom  11993  xrmaxadd  12005