xrpnfdc - Intuitionistic Logic Explorer

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Theorem xrpnfdc 9984

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 9918	. 2 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8140	. . . . . 6 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2398	. . . . 5 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4	3	olcd 736	. . . 4 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
5		df-dc 837	. . . 4 ⊢ (DECID 𝐴 = +∞ ↔ (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
6	4, 5	sylibr 134	. . 3 ⊢ (𝐴 ∈ ℝ → DECID 𝐴 = +∞)
7		orc 714	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
8	7, 5	sylibr 134	. . 3 ⊢ (𝐴 = +∞ → DECID 𝐴 = +∞)
9		mnfnepnf 8148	. . . . . . 7 ⊢ -∞ ≠ +∞
10	9	neii 2379	. . . . . 6 ⊢ ¬ -∞ = +∞
11		eqeq1 2213	. . . . . 6 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
12	10, 11	mtbiri 677	. . . . 5 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
13	12	olcd 736	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ∨ ¬ 𝐴 = +∞))
14	13, 5	sylibr 134	. . 3 ⊢ (𝐴 = -∞ → DECID 𝐴 = +∞)
15	6, 8, 14	3jaoi 1316	. 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 710 DECID wdc 836 ∨ w3o 980 = wceq 1373 ∈ wcel 2177 ℝcr 7944 +∞cpnf 8124 -∞cmnf 8125 ℝ^*cxr 8126

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 711 ax-5 1471 ax-7 1472 ax-gen 1473 ax-ie1 1517 ax-ie2 1518 ax-8 1528 ax-10 1529 ax-11 1530 ax-i12 1531 ax-bndl 1533 ax-4 1534 ax-17 1550 ax-i9 1554 ax-ial 1558 ax-i5r 1559 ax-13 2179 ax-14 2180 ax-ext 2188 ax-sep 4170 ax-pow 4226 ax-un 4488 ax-cnex 8036 ax-resscn 8037

This theorem depends on definitions: df-bi 117 df-dc 837 df-3or 982 df-3an 983 df-tru 1376 df-fal 1379 df-nf 1485 df-sb 1787 df-clab 2193 df-cleq 2199 df-clel 2202 df-nfc 2338 df-ne 2378 df-nel 2473 df-rex 2491 df-rab 2494 df-v 2775 df-un 3174 df-in 3176 df-ss 3183 df-pw 3623 df-sn 3644 df-pr 3645 df-uni 3857 df-pnf 8129 df-mnf 8130 df-xr 8131

This theorem is referenced by: xaddf 9986 xaddval 9987 xaddpnf1 9988 xaddcom 10003 xnegdi 10010 xleadd1a 10015 xlesubadd 10025 xrmaxiflemcl 11631 xrmaxifle 11632 xrmaxiflemab 11633 xrmaxiflemlub 11634 xrmaxiflemcom 11635 xrmaxadd 11647 xblss2ps 14951 xblss2 14952