xrrebnd - Metamath Proof Explorer

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Theorem xrrebnd 12831

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 12788	. . 3 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
2		ltpnf 12785	. . 3 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
3	1, 2	jca 511	. 2 ⊢ (𝐴 ∈ ℝ → (-∞ < 𝐴 ∧ 𝐴 < +∞))
4		nltpnft 12827	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
5		ngtmnft 12829	. . . . . 6 ⊢ (𝐴 ∈ ℝ^* → (𝐴 = -∞ ↔ ¬ -∞ < 𝐴))
6	4, 5	orbi12d 915	. . . . 5 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴)))
7		ianor 978	. . . . . 6 ⊢ (¬ (-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ (¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞))
8		orcom 866	. . . . . 6 ⊢ ((¬ -∞ < 𝐴 ∨ ¬ 𝐴 < +∞) ↔ (¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴))
9	7, 8	bitr2i 275	. . . . 5 ⊢ ((¬ 𝐴 < +∞ ∨ ¬ -∞ < 𝐴) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞))
10	6, 9	bitrdi 286	. . . 4 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ¬ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
11	10	con2bid 354	. . 3 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) ↔ ¬ (𝐴 = +∞ ∨ 𝐴 = -∞)))
12		elxr 12781	. . . . 5 ⊢ (𝐴 ∈ ℝ^* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
13		3orass 1088	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ (𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞)))
14		orcom 866	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∨ (𝐴 = +∞ ∨ 𝐴 = -∞)) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
15	13, 14	bitri 274	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ↔ ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
16	12, 15	sylbb 218	. . . 4 ⊢ (𝐴 ∈ ℝ^* → ((𝐴 = +∞ ∨ 𝐴 = -∞) ∨ 𝐴 ∈ ℝ))
17	16	ord 860	. . 3 ⊢ (𝐴 ∈ ℝ^* → (¬ (𝐴 = +∞ ∨ 𝐴 = -∞) → 𝐴 ∈ ℝ))
18	11, 17	sylbid 239	. 2 ⊢ (𝐴 ∈ ℝ^* → ((-∞ < 𝐴 ∧ 𝐴 < +∞) → 𝐴 ∈ ℝ))
19	3, 18	impbid2 225	1 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))

Colors of variables: wff setvar class

Syntax hints: ¬ wn 3 → wi 4 ↔ wb 205 ∧ wa 395 ∨ wo 843 ∨ w3o 1084 = wceq 1539 ∈ wcel 2108 class class class wbr 5070 ℝcr 10801 +∞cpnf 10937 -∞cmnf 10938 ℝ^*cxr 10939 < clt 10940

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1799 ax-4 1813 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2110 ax-9 2118 ax-10 2139 ax-11 2156 ax-12 2173 ax-ext 2709 ax-sep 5218 ax-nul 5225 ax-pow 5283 ax-pr 5347 ax-un 7566 ax-cnex 10858 ax-resscn 10859 ax-pre-lttri 10876 ax-pre-lttrn 10877

This theorem depends on definitions: df-bi 206 df-an 396 df-or 844 df-3or 1086 df-3an 1087 df-tru 1542 df-fal 1552 df-ex 1784 df-nf 1788 df-sb 2069 df-mo 2540 df-eu 2569 df-clab 2716 df-cleq 2730 df-clel 2817 df-nfc 2888 df-ne 2943 df-nel 3049 df-ral 3068 df-rex 3069 df-rab 3072 df-v 3424 df-sbc 3712 df-csb 3829 df-dif 3886 df-un 3888 df-in 3890 df-ss 3900 df-nul 4254 df-if 4457 df-pw 4532 df-sn 4559 df-pr 4561 df-op 4565 df-uni 4837 df-br 5071 df-opab 5133 df-mpt 5154 df-id 5480 df-po 5494 df-so 5495 df-xp 5586 df-rel 5587 df-cnv 5588 df-co 5589 df-dm 5590 df-rn 5591 df-res 5592 df-ima 5593 df-iota 6376 df-fun 6420 df-fn 6421 df-f 6422 df-f1 6423 df-fo 6424 df-f1o 6425 df-fv 6426 df-er 8456 df-en 8692 df-dom 8693 df-sdom 8694 df-pnf 10942 df-mnf 10943 df-xr 10944 df-ltxr 10945 df-le 10946

This theorem is referenced by: xrre 12832 xrre2 12833 xrre3 12834 supxrre1 12993 elioc2 13071 elico2 13072 elicc2 13073 xblpnfps 23456 xblpnf 23457 isnghm3 23795 ovoliun 24574 ovolicopnf 24593 voliunlem3 24621 volsup 24625 itg2seq 24812 nmblore 29049 nmopre 30133 supxrgere 42762 supxrgelem 42766 supxrge 42767 suplesup 42768 infrpge 42780 limsupre 43072

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