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Theorem xrre 8833

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 491	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → -∞ < 𝐴)
2		ltpnf 8802	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
3	2	adantl 266	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
4		rexr 7129	. . . . . 6 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
5		pnfxr 8792	. . . . . . 7 ⊢ +∞ ∈ ℝ^*
6		xrlelttr 8822	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ +∞ ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
7	5, 6	mp3an3 1232	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ^*) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
8	4, 7	sylan2 274	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((𝐴 ≤ 𝐵 ∧ 𝐵 < +∞) → 𝐴 < +∞))
9	3, 8	mpan2d 412	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐴 ≤ 𝐵 → 𝐴 < +∞))
10	9	imp 119	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐴 ≤ 𝐵) → 𝐴 < +∞)
11	10	adantrl 455	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 < +∞)
12		xrrebnd 8832	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
13	12	ad2antrr 465	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
14	1, 11, 13	mpbir2and 862	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (-∞ < 𝐴 ∧ 𝐴 ≤ 𝐵)) → 𝐴 ∈ ℝ)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 101 ↔ wb 102 ∈ wcel 1409 class class class wbr 3791 ℝcr 6945 +∞cpnf 7115 -∞cmnf 7116 ℝ^*cxr 7117 < clt 7118 ≤ cle 7119

This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 103 ax-ia2 104 ax-ia3 105 ax-in1 554 ax-in2 555 ax-io 640 ax-5 1352 ax-7 1353 ax-gen 1354 ax-ie1 1398 ax-ie2 1399 ax-8 1411 ax-10 1412 ax-11 1413 ax-i12 1414 ax-bndl 1415 ax-4 1416 ax-13 1420 ax-14 1421 ax-17 1435 ax-i9 1439 ax-ial 1443 ax-i5r 1444 ax-ext 2038 ax-sep 3902 ax-pow 3954 ax-pr 3971 ax-un 4197 ax-setind 4289 ax-cnex 7032 ax-resscn 7033 ax-pre-ltirr 7053 ax-pre-ltwlin 7054 ax-pre-lttrn 7055

This theorem depends on definitions: df-bi 114 df-3or 897 df-3an 898 df-tru 1262 df-fal 1265 df-nf 1366 df-sb 1662 df-eu 1919 df-mo 1920 df-clab 2043 df-cleq 2049 df-clel 2052 df-nfc 2183 df-ne 2221 df-nel 2315 df-ral 2328 df-rex 2329 df-rab 2332 df-v 2576 df-dif 2947 df-un 2949 df-in 2951 df-ss 2958 df-pw 3388 df-sn 3408 df-pr 3409 df-op 3411 df-uni 3608 df-br 3792 df-opab 3846 df-po 4060 df-iso 4061 df-xp 4378 df-cnv 4380 df-pnf 7120 df-mnf 7121 df-xr 7122 df-ltxr 7123 df-le 7124

This theorem is referenced by: xrrege0 8838

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