xrre4 - Mathbox for Glauco Siliprandi

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Theorem xrre4 44765

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 11287	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ -∞)
2	1	adantl 481	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐴 ∈ ℝ) → 𝐴 ≠ -∞)
3		renepnf 11286	. . . . 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 770	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ -∞)
9		simprr 772	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ≠ +∞)
10	7, 8, 9	xrred 44719	. . 3 ⊢ ((𝐴 ∈ ℝ^* ∧ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)) → 𝐴 ∈ ℝ)
11	10	ex 412	. 2 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞) → 𝐴 ∈ ℝ))
12	6, 11	impbid 211	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (𝐴 ≠ -∞ ∧ 𝐴 ≠ +∞)))

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 ∈ wcel 2099 ≠ wne 2935 ℝcr 11131 +∞cpnf 11269 -∞cmnf 11270 ℝ^*cxr 11271

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2164 ax-ext 2698 ax-sep 5293 ax-nul 5300 ax-pow 5359 ax-pr 5423 ax-un 7734 ax-cnex 11188 ax-resscn 11189

This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2529 df-eu 2558 df-clab 2705 df-cleq 2719 df-clel 2805 df-nfc 2880 df-ne 2936 df-nel 3042 df-ral 3057 df-rex 3066 df-rab 3428 df-v 3471 df-sbc 3775 df-csb 3890 df-dif 3947 df-un 3949 df-in 3951 df-ss 3961 df-nul 4319 df-if 4525 df-pw 4600 df-sn 4625 df-pr 4627 df-op 4631 df-uni 4904 df-br 5143 df-opab 5205 df-mpt 5226 df-id 5570 df-xp 5678 df-rel 5679 df-cnv 5680 df-co 5681 df-dm 5682 df-rn 5683 df-res 5684 df-ima 5685 df-iota 6494 df-fun 6544 df-fn 6545 df-f 6546 df-f1 6547 df-fo 6548 df-f1o 6549 df-fv 6550 df-er 8718 df-en 8958 df-dom 8959 df-sdom 8960 df-pnf 11274 df-mnf 11275 df-xr 11276

This theorem is referenced by: limsupre2lem 45084