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Theorem xrre3 10203

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 10164	. . . . . 6 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
2	1	adantl 277	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
3		mnfxr 8372	. . . . . . 7 ⊢ -∞ ∈ ℝ^*
4	3	a1i 9	. . . . . 6 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ^*)
5		rexr 8361	. . . . . . 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 10188	. . . . . 6 ⊢ ((-∞ ∈ ℝ^* ∧ 𝐵 ∈ ℝ^* ∧ 𝐴 ∈ ℝ^*) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴))
9	4, 6, 7, 8	syl3anc 1278	. . . . 5 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → ((-∞ < 𝐵 ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴))
10	2, 9	mpand 433	. . . 4 ⊢ ((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) → (𝐵 ≤ 𝐴 → -∞ < 𝐴))
11	10	imp 124	. . 3 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ 𝐵 ≤ 𝐴) → -∞ < 𝐴)
12	11	adantrr 483	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → -∞ < 𝐴)
13		simprr 537	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → 𝐴 < +∞)
14		xrrebnd 10200	. . 3 ⊢ (𝐴 ∈ ℝ^* → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
15	14	ad2antrr 492	. 2 ⊢ (((𝐴 ∈ ℝ^* ∧ 𝐵 ∈ ℝ) ∧ (𝐵 ≤ 𝐴 ∧ 𝐴 < +∞)) → (𝐴 ∈ ℝ ↔ (-∞ < 𝐴 ∧ 𝐴 < +∞)))
16	12, 13, 15	mpbir2and 957	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 2209 class class class wbr 4125 ℝcr 8168 +∞cpnf 8347 -∞cmnf 8348 ℝ^*cxr 8349 < clt 8350 ≤ cle 8351

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 623 ax-in2 624 ax-io 721 ax-5 1500 ax-7 1501 ax-gen 1502 ax-ie1 1546 ax-ie2 1547 ax-8 1557 ax-10 1558 ax-11 1559 ax-i12 1560 ax-bndl 1562 ax-4 1563 ax-17 1579 ax-i9 1583 ax-ial 1587 ax-i5r 1588 ax-14 2212 ax-ext 2220 ax-sep 4244 ax-pow 4306 ax-pr 4341 ax-un 4573 ax-setind 4679 ax-cnex 8260 ax-resscn 8261 ax-pre-ltirr 8281 ax-pre-ltwlin 8282 ax-pre-lttrn 8283

This theorem depends on definitions: df-bi 117 df-3or 1010 df-3an 1011 df-tru 1405 df-fal 1408 df-nf 1514 df-sb 1816 df-eu 2089 df-mo 2090 df-clab 2225 df-cleq 2231 df-clel 2234 df-nfc 2381 df-ne 2421 df-nel 2516 df-ral 2533 df-rex 2534 df-rab 2537 df-v 2823 df-dif 3222 df-un 3224 df-in 3226 df-ss 3233 df-pw 3687 df-sn 3711 df-pr 3712 df-op 3714 df-uni 3931 df-br 4126 df-opab 4188 df-po 4436 df-iso 4437 df-xp 4775 df-cnv 4777 df-pnf 8352 df-mnf 8353 df-xr 8354 df-ltxr 8355 df-le 8356

This theorem is referenced by: elicore 10679

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