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Theorem xrrebnd 11873

Description: An extended real is real iff it is strictly bounded by infinities. (Contributed by NM, 2-Feb-2006.)

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

**Proof of Theorem xrrebnd**
Step	Hyp	Ref	Expression
1		mnflt 11833	. . 3 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
2		ltpnf 11830	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
3	1, 2	jca 553	. 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ∧ 𝐴 < +∞))
4		nltpnft 11871	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
5		ngtmnft 11872	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
6	4, 5	orbi12d 742	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴)))
7		ianor 508	. . . . . 6 ⊢ (¬ (-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ (¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞))
8		orcom 401	. . . . . 6 ⊢ ((¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴))
9	7, 8	bitr2i 264	. . . . 5 ⊢ ((¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞))
10	6, 9	syl6bb 275	. . . 4 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
11	10	con2bid 343	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ ¬ (𝐴 = +∞ ∨ 𝐴 = -∞)))
12		elxr 11826	. . . . 5 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
13		3orass 1034	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞)))
14		orcom 401	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞)) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
15	13, 14	bitri 263	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
16	12, 15	sylbb 208	. . . 4 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
17	16	ord 391	. . 3 ⊢ (𝐴 ∈ ℝ^* → (¬ (𝐴 = +∞ ∨ 𝐴 = -∞) → 𝐴 ∈ ℝ))
18	11, 17	sylbid 229	. 2 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) → 𝐴 ∈ ℝ))
19	3, 18	impbid2 215	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 195 ∨ wo 382 ∧ wa 383 ∨ w3o 1030 = wceq 1475 ∈ wcel 1977 class class class wbr 4583 ℝcr 9814 +∞cpnf 9950 -∞cmnf 9951 ℝ^*cxr 9952 < clt 9953

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1713 ax-4 1728 ax-5 1827 ax-6 1875 ax-7 1922 ax-8 1979 ax-9 1986 ax-10 2006 ax-11 2021 ax-12 2034 ax-13 2234 ax-ext 2590 ax-sep 4709 ax-nul 4717 ax-pow 4769 ax-pr 4833 ax-un 6847 ax-cnex 9871 ax-resscn 9872 ax-pre-lttri 9889 ax-pre-lttrn 9890

This theorem depends on definitions: df-bi 196 df-or 384 df-an 385 df-3or 1032 df-3an 1033 df-tru 1478 df-ex 1696 df-nf 1701 df-sb 1868 df-eu 2462 df-mo 2463 df-clab 2597 df-cleq 2603 df-clel 2606 df-nfc 2740 df-ne 2782 df-nel 2783 df-ral 2901 df-rex 2902 df-rab 2905 df-v 3175 df-sbc 3403 df-csb 3500 df-dif 3543 df-un 3545 df-in 3547 df-ss 3554 df-nul 3875 df-if 4037 df-pw 4110 df-sn 4126 df-pr 4128 df-op 4132 df-uni 4373 df-br 4584 df-opab 4644 df-mpt 4645 df-id 4953 df-po 4959 df-so 4960 df-xp 5044 df-rel 5045 df-cnv 5046 df-co 5047 df-dm 5048 df-rn 5049 df-res 5050 df-ima 5051 df-iota 5768 df-fun 5806 df-fn 5807 df-f 5808 df-f1 5809 df-fo 5810 df-f1o 5811 df-fv 5812 df-er 7629 df-en 7842 df-dom 7843 df-sdom 7844 df-pnf 9955 df-mnf 9956 df-xr 9957 df-ltxr 9958 df-le 9959

This theorem is referenced by: xrre 11874 xrre2 11875 xrre3 11876 supxrre1 12032 elioc2 12107 elico2 12108 elicc2 12109 xblpnfps 22010 xblpnf 22011 isnghm3 22339 ovoliun 23080 ovolicopnf 23099 voliunlem3 23127 volsup 23131 itg2seq 23315 nmblore 27025 nmopre 28113 supxrgere 38490 supxrgelem 38494 supxrge 38495 suplesup 38496 infrpge 38508 limsupre 38708

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