xrpnfdc - Intuitionistic Logic Explorer

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Theorem xrpnfdc 9911

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 9845	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8069	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2385	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4	3	olcd 735	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5		df-dc 836	. . . 4 ⊢ (DECID 𝐴 = +∞ ↔ (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
6	4, 5	sylibr 134	. . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = +∞)
7		orc 713	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
8	7, 5	sylibr 134	. . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = +∞)
9		mnfnepnf 8077	. . . . . . 7 ⊢ -∞ ≠ +∞
10	9	neii 2366	. . . . . 6 ⊢ ¬ -∞ = +∞
11		eqeq1 2200	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
12	10, 11	mtbiri 676	. . . . 5 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
13	12	olcd 735	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
14	13, 5	sylibr 134	. . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = +∞)
15	6, 8, 14	3jaoi 1314	. 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 709 DECID wdc 835 ∨ w3o 979 = wceq 1364 ∈ wcel 2164 ℝcr 7873 +∞cpnf 8053 -∞cmnf 8054 ℝ^*cxr 8055

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 615 ax-in2 616 ax-io 710 ax-5 1458 ax-7 1459 ax-gen 1460 ax-ie1 1504 ax-ie2 1505 ax-8 1515 ax-10 1516 ax-11 1517 ax-i12 1518 ax-bndl 1520 ax-4 1521 ax-17 1537 ax-i9 1541 ax-ial 1545 ax-i5r 1546 ax-13 2166 ax-14 2167 ax-ext 2175 ax-sep 4148 ax-pow 4204 ax-un 4465 ax-cnex 7965 ax-resscn 7966

This theorem depends on definitions: df-bi 117 df-dc 836 df-3or 981 df-3an 982 df-tru 1367 df-fal 1370 df-nf 1472 df-sb 1774 df-clab 2180 df-cleq 2186 df-clel 2189 df-nfc 2325 df-ne 2365 df-nel 2460 df-rex 2478 df-rab 2481 df-v 2762 df-un 3158 df-in 3160 df-ss 3167 df-pw 3604 df-sn 3625 df-pr 3626 df-uni 3837 df-pnf 8058 df-mnf 8059 df-xr 8060

This theorem is referenced by: xaddf 9913 xaddval 9914 xaddpnf1 9915 xaddcom 9930 xnegdi 9937 xleadd1a 9942 xlesubadd 9952 xrmaxiflemcl 11391 xrmaxifle 11392 xrmaxiflemab 11393 xrmaxiflemlub 11394 xrmaxiflemcom 11395 xrmaxadd 11407 xblss2ps 14583 xblss2 14584