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Theorem xrre 12373

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 758	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴)
2		ltpnf 12326	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
3	2	adantl 474	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
4		rexr 10480	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
5		pnfxr 10488	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6		xrlelttr 12360	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
7	5, 6	mp3an3 1429	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
8	4, 7	sylan2 583	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
9	3, 8	mpan2d 681	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 < +∞))
10	9	imp 398	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → 𝐴 < +∞)
11	10	adantrl 703	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 < +∞)
12		xrrebnd 12372	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
13	12	ad2antrr 713	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
14	1, 11, 13	mpbir2and 700	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

Syntax hints: → wi 4 ↔ wb 198 ∧ wa 387 ∈ wcel 2050 class class class wbr 4923 ℝcr 10328 +∞cpnf 10465 -∞cmnf 10466 ℝ^*cxr 10467 < clt 10468 ≤ cle 10469

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-13 2301 ax-ext 2744 ax-sep 5054 ax-nul 5061 ax-pow 5113 ax-pr 5180 ax-un 7273 ax-cnex 10385 ax-resscn 10386 ax-pre-lttri 10403 ax-pre-lttrn 10404

This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-3or 1069 df-3an 1070 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-mo 2547 df-eu 2584 df-clab 2753 df-cleq 2765 df-clel 2840 df-nfc 2912 df-ne 2962 df-nel 3068 df-ral 3087 df-rex 3088 df-rab 3091 df-v 3411 df-sbc 3676 df-csb 3781 df-dif 3826 df-un 3828 df-in 3830 df-ss 3837 df-nul 4173 df-if 4345 df-pw 4418 df-sn 4436 df-pr 4438 df-op 4442 df-uni 4707 df-br 4924 df-opab 4986 df-mpt 5003 df-id 5306 df-po 5320 df-so 5321 df-xp 5407 df-rel 5408 df-cnv 5409 df-co 5410 df-dm 5411 df-rn 5412 df-res 5413 df-ima 5414 df-iota 6146 df-fun 6184 df-fn 6185 df-f 6186 df-f1 6187 df-fo 6188 df-f1o 6189 df-fv 6190 df-er 8083 df-en 8301 df-dom 8302 df-sdom 8303 df-pnf 10470 df-mnf 10471 df-xr 10472 df-ltxr 10473 df-le 10474

This theorem is referenced by: xrrege0 12378 supxrre 12530 infxrre 12539 caucvgrlem 14884 pcgcd1 16063 tgioo 23101 ovolunlem1a 23794 ovoliunlem1 23800 ioombl1lem2 23857 itg2monolem2 24049 dvferm1lem 24278 radcnvle 24705 psercnlem1 24710 nmobndi 28323 ubthlem3 28421 nmophmi 29583 bdophsi 29648 bdopcoi 29650 orvclteel 31376 itg2addnclem 34384 itg2gt0cn 34388 areacirclem5 34427 eliocre 41216 fourierdlem87 41909 sge0ssre 42110

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