xrre4 - Mathbox for Glauco Siliprandi

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Theorem xrre4 46421

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 11358	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
2	1	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
3		renepnf 11357	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
4	3	adantl 487	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞)
5	2, 4	jca 521	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))
6	5	ex 418	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))
7		simpl 488	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ^*)
8		simprl 783	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ -∞)
9		simprr 785	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ +∞)
10	7, 8, 9	xrred 46376	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ)
11	10	ex 418	. 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

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 401 ∈ wcel 2144 ≠ wne 2955 ℝcr 11199 +∞cpnf 11340 -∞cmnf 11341 ℝ^*cxr 11342

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1827 ax-4 1841 ax-5 1942 ax-6 1999 ax-7 2040 ax-8 2146 ax-9 2154 ax-10 2177 ax-11 2193 ax-12 2212 ax-ext 2732 ax-sep 5248 ax-nul 5259 ax-pow 5326 ax-pr 5390 ax-un 7751 ax-cnex 11256 ax-resscn 11257

This proof depends on definitions: df-bi 210 df-an 402 df-or 862 df-3or 1104 df-3an 1105 df-tru 1572 df-fal 1582 df-ex 1812 df-nf 1816 df-sb 2099 df-mo 2564 df-eu 2594 df-clab 2739 df-cleq 2752 df-clel 2835 df-nfc 2909 df-ne 2956 df-nel 3062 df-ral 3077 df-rex 3087 df-rab 3413 df-v 3452 df-sbc 3739 df-csb 3847 df-ss 3896 df-dif 4047 df-in 4091 df-un 4151 df-nul 4279 df-if 4482 df-pw 4558 df-sn 4584 df-pr 4586 df-op 4590 df-uni 4867 df-br 5103 df-opab 5167 df-mpt 5186 df-id 5545 df-xp 5656 df-rel 5657 df-cnv 5658 df-co 5659 df-dm 5660 df-rn 5661 df-res 5662 df-ima 5663 df-iota 6494 df-fun 6540 df-fn 6541 df-f 6542 df-f1 6543 df-fo 6544 df-f1o 6545 df-fv 6546 df-er 8717 df-en 8974 df-dom 8975 df-sdom 8976 df-pnf 11345 df-mnf 11346 df-xr 11347

This theorem is used by: limsupre2lem 46734