xrpnfdc - Intuitionistic Logic Explorer

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Theorem xrpnfdc 9799

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 9733	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 7967	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2361	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4	3	olcd 729	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5		df-dc 830	. . . 4 ⊢ (DECID 𝐴 = +∞ ↔ (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
6	4, 5	sylibr 133	. . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = +∞)
7		orc 707	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
8	7, 5	sylibr 133	. . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = +∞)
9		mnfnepnf 7975	. . . . . . 7 ⊢ -∞ ≠ +∞
10	9	neii 2342	. . . . . 6 ⊢ ¬ -∞ = +∞
11		eqeq1 2177	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
12	10, 11	mtbiri 670	. . . . 5 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
13	12	olcd 729	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
14	13, 5	sylibr 133	. . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = +∞)
15	6, 8, 14	3jaoi 1298	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → DECID 𝐴 = +∞)
16	1, 15	sylbi 120	1 ⊢ (𝐴 ∈ ℝ^* → DECID 𝐴 = +∞)

Colors of variables: wff set class

Syntax hints: ¬ wn 3 → wi 4 ∨ wo 703 DECID wdc 829 ∨ w3o 972 = wceq 1348 ∈ wcel 2141 ℝcr 7773 +∞cpnf 7951 -∞cmnf 7952 ℝ^*cxr 7953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-in1 609 ax-in2 610 ax-io 704 ax-5 1440 ax-7 1441 ax-gen 1442 ax-ie1 1486 ax-ie2 1487 ax-8 1497 ax-10 1498 ax-11 1499 ax-i12 1500 ax-bndl 1502 ax-4 1503 ax-17 1519 ax-i9 1523 ax-ial 1527 ax-i5r 1528 ax-13 2143 ax-14 2144 ax-ext 2152 ax-sep 4107 ax-pow 4160 ax-un 4418 ax-cnex 7865 ax-resscn 7866

This theorem depends on definitions: df-bi 116 df-dc 830 df-3or 974 df-3an 975 df-tru 1351 df-fal 1354 df-nf 1454 df-sb 1756 df-clab 2157 df-cleq 2163 df-clel 2166 df-nfc 2301 df-ne 2341 df-nel 2436 df-rex 2454 df-rab 2457 df-v 2732 df-un 3125 df-in 3127 df-ss 3134 df-pw 3568 df-sn 3589 df-pr 3590 df-uni 3797 df-pnf 7956 df-mnf 7957 df-xr 7958

This theorem is referenced by: xaddf 9801 xaddval 9802 xaddpnf1 9803 xaddcom 9818 xnegdi 9825 xleadd1a 9830 xlesubadd 9840 xrmaxiflemcl 11208 xrmaxifle 11209 xrmaxiflemab 11210 xrmaxiflemlub 11211 xrmaxiflemcom 11212 xrmaxadd 11224 xblss2ps 13198 xblss2 13199