xrrebnd - Metamath Proof Explorer

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Theorem xrrebnd 13184

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 13139	. . 3 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
2		ltpnf 13136	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
3	1, 2	jca 511	. 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ∧ 𝐴 < +∞))
4		nltpnft 13180	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
5		ngtmnft 13182	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
6	4, 5	orbi12d 918	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴)))
7		ianor 983	. . . . . 6 ⊢ (¬ (-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ (¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞))
8		orcom 870	. . . . . 6 ⊢ ((¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴))
9	7, 8	bitr2i 276	. . . . 5 ⊢ ((¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞))
10	6, 9	bitrdi 287	. . . 4 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
11	10	con2bid 354	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ ¬ (𝐴 = +∞ ∨ 𝐴 = -∞)))
12		elxr 13132	. . . . 5 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
13		3orass 1089	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞)))
14		orcom 870	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞)) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
15	13, 14	bitri 275	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
16	12, 15	sylbb 219	. . . 4 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
17	16	ord 864	. . 3 ⊢ (𝐴 ∈ ℝ^* → (¬ (𝐴 = +∞ ∨ 𝐴 = -∞) → 𝐴 ∈ ℝ))
18	11, 17	sylbid 240	. 2 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) → 𝐴 ∈ ℝ))
19	3, 18	impbid2 226	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 206 ∧ wa 395 ∨ wo 847 ∨ w3o 1085 = wceq 1540 ∈ wcel 2108 class class class wbr 5119 ℝcr 11128 +∞cpnf 11266 -∞cmnf 11267 ℝ^*cxr 11268 < clt 11269

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1910 ax-6 1967 ax-7 2007 ax-8 2110 ax-9 2118 ax-10 2141 ax-11 2157 ax-12 2177 ax-ext 2707 ax-sep 5266 ax-nul 5276 ax-pow 5335 ax-pr 5402 ax-un 7729 ax-cnex 11185 ax-resscn 11186 ax-pre-lttri 11203 ax-pre-lttrn 11204

This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-3an 1088 df-tru 1543 df-fal 1553 df-ex 1780 df-nf 1784 df-sb 2065 df-mo 2539 df-eu 2568 df-clab 2714 df-cleq 2727 df-clel 2809 df-nfc 2885 df-ne 2933 df-nel 3037 df-ral 3052 df-rex 3061 df-rab 3416 df-v 3461 df-sbc 3766 df-csb 3875 df-dif 3929 df-un 3931 df-in 3933 df-ss 3943 df-nul 4309 df-if 4501 df-pw 4577 df-sn 4602 df-pr 4604 df-op 4608 df-uni 4884 df-br 5120 df-opab 5182 df-mpt 5202 df-id 5548 df-po 5561 df-so 5562 df-xp 5660 df-rel 5661 df-cnv 5662 df-co 5663 df-dm 5664 df-rn 5665 df-res 5666 df-ima 5667 df-iota 6484 df-fun 6533 df-fn 6534 df-f 6535 df-f1 6536 df-fo 6537 df-f1o 6538 df-fv 6539 df-er 8719 df-en 8960 df-dom 8961 df-sdom 8962 df-pnf 11271 df-mnf 11272 df-xr 11273 df-ltxr 11274 df-le 11275

This theorem is referenced by: xrre 13185 xrre2 13186 xrre3 13187 supxrre1 13346 elioc2 13426 elico2 13427 elicc2 13428 xblpnfps 24334 xblpnf 24335 isnghm3 24664 ovoliun 25458 ovolicopnf 25477 voliunlem3 25505 volsup 25509 itg2seq 25695 nmblore 30767 nmopre 31851 supxrgere 45360 supxrgelem 45364 supxrge 45365 suplesup 45366 infrpge 45378 limsupre 45670

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