xrpnfdc - Intuitionistic Logic Explorer

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Theorem xrpnfdc 10181

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

**Assertion**
Ref	Expression
xrpnfdc	⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = +∞)

**Proof of Theorem xrpnfdc**
Step	Hyp	Ref	Expression
1		elxr 10115	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8326	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2435	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4	3	olcd 742	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5		df-dc 843	. . . 4 ⊢ (DECID 𝐴 = +∞ ↔ (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
6	4, 5	sylibr 134	. . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = +∞)
7		orc 720	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
8	7, 5	sylibr 134	. . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = +∞)
9		mnfnepnf 8334	. . . . . . 7 ⊢ -∞ ≠ +∞
10	9	neii 2416	. . . . . 6 ⊢ ¬ -∞ = +∞
11		eqeq1 2241	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
12	10, 11	mtbiri 682	. . . . 5 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
13	12	olcd 742	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
14	13, 5	sylibr 134	. . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = +∞)
15	6, 8, 14	3jaoi 1340	. 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 716 DECID wdc 842 ∨ w3o 1004 = wceq 1398 ∈ wcel 2205 ℝcr 8131 +∞cpnf 8310 -∞cmnf 8311 ℝ^*cxr 8312

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 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4230 ax-pow 4289 ax-un 4556 ax-cnex 8223 ax-resscn 8224

This theorem depends on definitions: df-bi 117 df-dc 843 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-rex 2528 df-rab 2531 df-v 2817 df-un 3217 df-in 3219 df-ss 3226 df-pw 3673 df-sn 3697 df-pr 3698 df-uni 3917 df-pnf 8315 df-mnf 8316 df-xr 8317

This theorem is referenced by: xaddf 10183 xaddval 10184 xaddpnf1 10185 xaddcom 10200 xnegdi 10207 xleadd1a 10212 xlesubadd 10222 xrmaxiflemcl 11938 xrmaxifle 11939 xrmaxiflemab 11940 xrmaxiflemlub 11941 xrmaxiflemcom 11942 xrmaxadd 11954 xblss2ps 15318 xblss2 15319