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Theorem xrmnfdc 10228

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

**Assertion**
Ref	Expression
xrmnfdc	⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)

**Proof of Theorem xrmnfdc**
Step	Hyp	Ref	Expression
1		elxr 10161	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renemnf 8368	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
3	2	neneqd 2441	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = -∞)
4	3	olcd 746	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
5		df-dc 847	. . . 4 ⊢ (DECID 𝐴 = -∞ ↔ (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
6	4, 5	sylibr 134	. . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = -∞)
7		pnfnemnf 8374	. . . . . . 7 ⊢ +∞ ≠ -∞
8	7	neii 2422	. . . . . 6 ⊢ ¬ +∞ = -∞
9		eqeq1 2245	. . . . . 6 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ↔ +∞ = -∞))
10	8, 9	mtbiri 686	. . . . 5 ⊢ (𝐴 = +∞ → ¬ 𝐴 = -∞)
11	10	olcd 746	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
12	11, 5	sylibr 134	. . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = -∞)
13		orc 724	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 = -∞ ∨ ¬ 𝐴 = -∞))
14	13, 5	sylibr 134	. . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = -∞)
15	6, 12, 14	3jaoi 1344	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → DECID 𝐴 = -∞)
16	1, 15	sylbi 121	1 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = -∞)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 720 DECID wdc 846 ∨ w3o 1008 = wceq 1402 ∈ wcel 2209 ℝcr 8172 +∞cpnf 8351 -∞cmnf 8352 ℝ^*cxr 8353

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 4247 ax-pow 4309 ax-un 4576 ax-setind 4682 ax-cnex 8264 ax-resscn 8265

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 3690 df-sn 3714 df-pr 3715 df-uni 3934 df-pnf 8356 df-mnf 8357 df-xr 8358

This theorem is referenced by: xaddf 10229 xaddval 10230 xaddmnf1 10233 xaddcom 10246 xnegdi 10253 xpncan 10256 xleadd1a 10258 xsubge0 10266 xrmaxiflemcl 11994 xrmaxifle 11995 xrmaxiflemab 11996 xrmaxiflemlub 11997 xrmaxiflemcom 11998 xrmaxadd 12010