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Theorem xrre3 10174

Description: A way of proving that an extended real is real. (Contributed by FL, 29-May-2014.)

**Assertion**
Ref	Expression
xrre3	⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ)

**Proof of Theorem xrre3**
Step	Hyp	Ref	Expression
1		mnflt 10135	. . . . . 6 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
2	1	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
3		mnfxr 8346	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
4	3	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ^*)
5		rexr 8335	. . . . . . 7 ⊢ (𝐵 ∈ ℝ → 𝐵 ∈ ℝ^*)
6	5	adantl 277	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ^*)
7		simpl 109	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ^*)
8		xrltletr 10159	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴))
9	4, 6, 7, 8	syl3anc 1274	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴))
10	2, 9	mpand 429	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 → -∞ < 𝐴))
11	10	imp 124	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴)
12	11	adantrr 479	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → -∞ < 𝐴)
13		simprr 533	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 < +∞)
14		xrrebnd 10171	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
15	14	ad2antrr 488	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
16	12, 13, 15	mpbir2and 953	1 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 ∈ ℝ)

Colors of variables: wff set class

Syntax hints: → wi 4 ∧ wa 104 ↔ wb 105 ∈ wcel 2205 class class class wbr 4114 ℝcr 8142 +∞cpnf 8321 -∞cmnf 8322 ℝ^*cxr 8323 < clt 8324 ≤ cle 8325

This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 106 ax-ia2 107 ax-ia3 108 ax-in1 619 ax-in2 620 ax-io 717 ax-5 1496 ax-7 1497 ax-gen 1498 ax-ie1 1542 ax-ie2 1543 ax-8 1553 ax-10 1554 ax-11 1555 ax-i12 1556 ax-bndl 1558 ax-4 1559 ax-17 1575 ax-i9 1579 ax-ial 1583 ax-i5r 1584 ax-13 2207 ax-14 2208 ax-ext 2216 ax-sep 4233 ax-pow 4292 ax-pr 4327 ax-un 4559 ax-setind 4664 ax-cnex 8234 ax-resscn 8235 ax-pre-ltirr 8255 ax-pre-ltwlin 8256 ax-pre-lttrn 8257

This theorem depends on definitions: df-bi 117 df-3or 1006 df-3an 1007 df-tru 1401 df-fal 1404 df-nf 1510 df-sb 1812 df-eu 2085 df-mo 2086 df-clab 2221 df-cleq 2227 df-clel 2230 df-nfc 2375 df-ne 2415 df-nel 2510 df-ral 2527 df-rex 2528 df-rab 2531 df-v 2817 df-dif 3216 df-un 3218 df-in 3220 df-ss 3227 df-pw 3676 df-sn 3700 df-pr 3701 df-op 3703 df-uni 3920 df-br 4115 df-opab 4177 df-po 4422 df-iso 4423 df-xp 4760 df-cnv 4762 df-pnf 8326 df-mnf 8327 df-xr 8328 df-ltxr 8329 df-le 8330

This theorem is referenced by: elicore 10650

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