xrpnfdc - Intuitionistic Logic Explorer

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Theorem xrpnfdc 10006

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 9940	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8162	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2401	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4	3	olcd 738	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5		df-dc 839	. . . 4 ⊢ (DECID 𝐴 = +∞ ↔ (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
6	4, 5	sylibr 134	. . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = +∞)
7		orc 716	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
8	7, 5	sylibr 134	. . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = +∞)
9		mnfnepnf 8170	. . . . . . 7 ⊢ -∞ ≠ +∞
10	9	neii 2382	. . . . . 6 ⊢ ¬ -∞ = +∞
11		eqeq1 2216	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
12	10, 11	mtbiri 679	. . . . 5 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
13	12	olcd 738	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
14	13, 5	sylibr 134	. . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = +∞)
15	6, 8, 14	3jaoi 1318	. 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 712 DECID wdc 838 ∨ w3o 982 = wceq 1375 ∈ wcel 2180 ℝcr 7966 +∞cpnf 8146 -∞cmnf 8147 ℝ^*cxr 8148

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 617 ax-in2 618 ax-io 713 ax-5 1473 ax-7 1474 ax-gen 1475 ax-ie1 1519 ax-ie2 1520 ax-8 1530 ax-10 1531 ax-11 1532 ax-i12 1533 ax-bndl 1535 ax-4 1536 ax-17 1552 ax-i9 1556 ax-ial 1560 ax-i5r 1561 ax-13 2182 ax-14 2183 ax-ext 2191 ax-sep 4181 ax-pow 4237 ax-un 4501 ax-cnex 8058 ax-resscn 8059

This theorem depends on definitions: df-bi 117 df-dc 839 df-3or 984 df-3an 985 df-tru 1378 df-fal 1381 df-nf 1487 df-sb 1789 df-clab 2196 df-cleq 2202 df-clel 2205 df-nfc 2341 df-ne 2381 df-nel 2476 df-rex 2494 df-rab 2497 df-v 2781 df-un 3181 df-in 3183 df-ss 3190 df-pw 3631 df-sn 3652 df-pr 3653 df-uni 3868 df-pnf 8151 df-mnf 8152 df-xr 8153

This theorem is referenced by: xaddf 10008 xaddval 10009 xaddpnf1 10010 xaddcom 10025 xnegdi 10032 xleadd1a 10037 xlesubadd 10047 xrmaxiflemcl 11722 xrmaxifle 11723 xrmaxiflemab 11724 xrmaxiflemlub 11725 xrmaxiflemcom 11726 xrmaxadd 11738 xblss2ps 15043 xblss2 15044