xrre4 - Mathbox for Glauco Siliprandi

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Theorem xrre4 45860

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 11188	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
2	1	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
3		renepnf 11187	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
4	3	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ +∞)
5	2, 4	jca 511	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞))
6	5	ex 412	. 2 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ → (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))
7		simpl 482	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ^*)
8		simprl 771	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ -∞)
9		simprr 773	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ +∞)
10	7, 8, 9	xrred 45815	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ)
11	10	ex 412	. 2 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ))
12	6, 11	impbid 212	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 206 ∧ wa 395 ∈ wcel 2114 ≠ wne 2933 ℝcr 11031 +∞cpnf 11170 -∞cmnf 11171 ℝ^*cxr 11172

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 1912 ax-6 1969 ax-7 2010 ax-8 2116 ax-9 2124 ax-10 2147 ax-11 2163 ax-12 2185 ax-ext 2709 ax-sep 5232 ax-nul 5242 ax-pow 5303 ax-pr 5371 ax-un 7683 ax-cnex 11088 ax-resscn 11089

This theorem depends on definitions: df-bi 207 df-an 396 df-or 849 df-3or 1088 df-3an 1089 df-tru 1545 df-fal 1555 df-ex 1782 df-nf 1786 df-sb 2069 df-mo 2540 df-eu 2570 df-clab 2716 df-cleq 2729 df-clel 2812 df-nfc 2886 df-ne 2934 df-nel 3038 df-ral 3053 df-rex 3063 df-rab 3391 df-v 3432 df-sbc 3730 df-csb 3839 df-dif 3893 df-un 3895 df-in 3897 df-ss 3907 df-nul 4275 df-if 4468 df-pw 4544 df-sn 4569 df-pr 4571 df-op 4575 df-uni 4852 df-br 5087 df-opab 5149 df-mpt 5168 df-id 5520 df-xp 5631 df-rel 5632 df-cnv 5633 df-co 5634 df-dm 5635 df-rn 5636 df-res 5637 df-ima 5638 df-iota 6449 df-fun 6495 df-fn 6496 df-f 6497 df-f1 6498 df-fo 6499 df-f1o 6500 df-fv 6501 df-er 8637 df-en 8888 df-dom 8889 df-sdom 8890 df-pnf 11175 df-mnf 11176 df-xr 11177

This theorem is referenced by: limsupre2lem 46173