xrre4 - Mathbox for Glauco Siliprandi

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Theorem xrre4 42414

Description: An extended real is real iff it is not an infinty. (Contributed by Glauco Siliprandi, 23-Oct-2021.)

**Assertion**
Ref	Expression
xrre4	⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))

**Proof of Theorem xrre4**
Step	Hyp	Ref	Expression
1		renemnf 10728	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
2	1	adantl 485	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
3		renepnf 10727	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
4	3	adantl 485	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞)
5	2, 4	jca 515	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))
6	5	ex 416	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))
7		simpl 486	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ^*)
8		simprl 770	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ -∞)
9		simprr 772	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ +∞)
10	7, 8, 9	xrred 42365	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ)
11	10	ex 416	. 2 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ))
12	6, 11	impbid 215	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 209 ∧ wa 399 ∈ wcel 2111 ≠ wne 2951 ℝcr 10574 +∞cpnf 10710 -∞cmnf 10711 ℝ^*cxr 10712

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2729 ax-sep 5169 ax-nul 5176 ax-pow 5234 ax-pr 5298 ax-un 7459 ax-cnex 10631 ax-resscn 10632

This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3or 1085 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2557 df-eu 2588 df-clab 2736 df-cleq 2750 df-clel 2830 df-nfc 2901 df-ne 2952 df-nel 3056 df-ral 3075 df-rex 3076 df-rab 3079 df-v 3411 df-sbc 3697 df-csb 3806 df-dif 3861 df-un 3863 df-in 3865 df-ss 3875 df-nul 4226 df-if 4421 df-pw 4496 df-sn 4523 df-pr 4525 df-op 4529 df-uni 4799 df-br 5033 df-opab 5095 df-mpt 5113 df-id 5430 df-xp 5530 df-rel 5531 df-cnv 5532 df-co 5533 df-dm 5534 df-rn 5535 df-res 5536 df-ima 5537 df-iota 6294 df-fun 6337 df-fn 6338 df-f 6339 df-f1 6340 df-fo 6341 df-f1o 6342 df-fv 6343 df-er 8299 df-en 8528 df-dom 8529 df-sdom 8530 df-pnf 10715 df-mnf 10716 df-xr 10717

This theorem is referenced by: limsupre2lem 42732