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Theorem xrre 12164

Description: A way of proving that an extended real is real. (Contributed by NM, 9-Mar-2006.)

**Assertion**
Ref	Expression
xrre	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

**Proof of Theorem xrre**
Step	Hyp	Ref	Expression
1		simprl 811	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴)
2		ltpnf 12118	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
3	2	adantl 473	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
4		rexr 10248	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
5		pnfxr 10255	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6		xrlelttr 12151	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
7	5, 6	mp3an3 1550	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
8	4, 7	sylan2 492	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
9	3, 8	mpan2d 712	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 < +∞))
10	9	imp 444	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → 𝐴 < +∞)
11	10	adantrl 754	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 < +∞)
12		xrrebnd 12163	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
13	12	ad2antrr 764	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
14	1, 11, 13	mpbir2and 995	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 196 ∧ wa 383 ∈ wcel 2127 class class class wbr 4792 ℝcr 10098 +∞cpnf 10234 -∞cmnf 10235 ℝ^*cxr 10236 < clt 10237 ≤ cle 10238

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1859 ax-4 1874 ax-5 1976 ax-6 2042 ax-7 2078 ax-8 2129 ax-9 2136 ax-10 2156 ax-11 2171 ax-12 2184 ax-13 2379 ax-ext 2728 ax-sep 4921 ax-nul 4929 ax-pow 4980 ax-pr 5043 ax-un 7102 ax-cnex 10155 ax-resscn 10156 ax-pre-lttri 10173 ax-pre-lttrn 10174

This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-3or 1073 df-3an 1074 df-tru 1623 df-ex 1842 df-nf 1847 df-sb 2035 df-eu 2599 df-mo 2600 df-clab 2735 df-cleq 2741 df-clel 2744 df-nfc 2879 df-ne 2921 df-nel 3024 df-ral 3043 df-rex 3044 df-rab 3047 df-v 3330 df-sbc 3565 df-csb 3663 df-dif 3706 df-un 3708 df-in 3710 df-ss 3717 df-nul 4047 df-if 4219 df-pw 4292 df-sn 4310 df-pr 4312 df-op 4316 df-uni 4577 df-br 4793 df-opab 4853 df-mpt 4870 df-id 5162 df-po 5175 df-so 5176 df-xp 5260 df-rel 5261 df-cnv 5262 df-co 5263 df-dm 5264 df-rn 5265 df-res 5266 df-ima 5267 df-iota 6000 df-fun 6039 df-fn 6040 df-f 6041 df-f1 6042 df-fo 6043 df-f1o 6044 df-fv 6045 df-er 7899 df-en 8110 df-dom 8111 df-sdom 8112 df-pnf 10239 df-mnf 10240 df-xr 10241 df-ltxr 10242 df-le 10243

This theorem is referenced by: xrrege0 12169 supxrre 12321 infxrre 12330 caucvgrlem 14573 pcgcd1 15754 tgioo 22771 ovolunlem1a 23435 ovoliunlem1 23441 ioombl1lem2 23498 itg2monolem2 23688 dvferm1lem 23917 radcnvle 24344 psercnlem1 24349 nmobndi 27910 ubthlem3 28008 nmophmi 29170 bdophsi 29235 bdopcoi 29237 orvclteel 30814 itg2addnclem 33743 itg2gt0cn 33747 areacirclem5 33786 eliocre 40206 fourierdlem87 40882 sge0ssre 41086

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