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Theorem xrre 13201

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 782	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴)
2		ltpnf 13151	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
3	2	adantl 486	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
4		rexr 11261	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
5		pnfxr 11269	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6		xrlelttr 13187	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
7	5, 6	mp3an3 1478	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
8	4, 7	sylan2 604	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
9	3, 8	mpan2d 706	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 < +∞))
10	9	imp 411	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → 𝐴 < +∞)
11	10	adantrl 728	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 < +∞)
12		xrrebnd 13200	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
13	12	ad2antrr 738	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
14	1, 11, 13	mpbir2and 725	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

Colors of variables: wff setvar class

This proof depends on syntax axioms: → wi 4 ↔ wb 209 ∧ wa 400 ∈ wcel 2142 class class class wbr 5108 ℝcr 11105 +∞cpnf 11246 -∞cmnf 11247 ℝ^*cxr 11248 < clt 11249 ≤ cle 11250

This proof depends on axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1824 ax-4 1838 ax-5 1939 ax-6 1996 ax-7 2037 ax-8 2144 ax-9 2152 ax-10 2175 ax-11 2191 ax-12 2212 ax-ext 2734 ax-sep 5256 ax-nul 5268 ax-pow 5335 ax-pr 5403 ax-un 7734 ax-cnex 11162 ax-resscn 11163 ax-pre-lttri 11180 ax-pre-lttrn 11181

This proof depends on definitions: df-bi 210 df-an 401 df-or 861 df-3or 1103 df-3an 1104 df-tru 1572 df-fal 1582 df-ex 1809 df-nf 1813 df-sb 2096 df-mo 2566 df-eu 2596 df-clab 2741 df-cleq 2754 df-clel 2837 df-nfc 2911 df-ne 2958 df-nel 3064 df-ral 3079 df-rex 3089 df-rab 3416 df-v 3456 df-sbc 3744 df-csb 3853 df-dif 3907 df-un 3909 df-in 3911 df-ss 3921 df-nul 4286 df-if 4487 df-pw 4563 df-sn 4589 df-pr 4591 df-op 4595 df-uni 4872 df-br 5109 df-opab 5173 df-mpt 5192 df-id 5555 df-po 5568 df-so 5569 df-xp 5666 df-rel 5667 df-cnv 5668 df-co 5669 df-dm 5670 df-rn 5671 df-res 5672 df-ima 5673 df-iota 6492 df-fun 6538 df-fn 6539 df-f 6540 df-f1 6541 df-fo 6542 df-f1o 6543 df-fv 6544 df-er 8692 df-en 8942 df-dom 8943 df-sdom 8944 df-pnf 11251 df-mnf 11252 df-xr 11253 df-ltxr 11254 df-le 11255

This theorem is used by: xrrege0 13206 supxrre 13359 infxrre 13369 caucvgrlem 15731 pcgcd1 16943 tgioo 24964 ovolunlem1a 25666 ovoliunlem1 25672 ioombl1lem2 25729 itg2monolem2 25921 dvferm1lem 26154 radcnvle 26594 psercnlem1 26599 nmobndi 31138 ubthlem3 31235 nmophmi 32394 bdophsi 32459 bdopcoi 32461 orvclteel 34872 itg2addnclem 38350 itg2gt0cn 38354 areacirclem5 38391 eliocre 46253 fourierdlem87 46935 sge0ssre 47139

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