Theorem nltpnft 9883

Description: An extended real is not less than plus infinity iff they are equal. (Contributed by NM, 30-Jan-2006.)

Assertion

Ref	Expression
nltpnft	⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Proof of Theorem nltpnft

Step	Hyp	Ref	Expression
1		elxr 9845	. 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8069	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2385	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4		ltpnf 9849	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
5		notnot 630	. . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
6	4, 5	syl 14	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
7	3, 6	2falsed 703	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8		id 19	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
9		pnfxr 8074	. . . . . 6 ⊢ +∞ ∈ ℝ*
10		xrltnr 9848	. . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ¬ +∞ < +∞
12		breq1 4033	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
13	11, 12	mtbiri 676	. . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
14	8, 13	2thd 175	. . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15		mnfnepnf 8077	. . . . . 6 ⊢ -∞ ≠ +∞
16	15	neii 2366	. . . . 5 ⊢ ¬ -∞ = +∞
17		eqeq1 2200	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
18	16, 17	mtbiri 676	. . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
19		mnfltpnf 9854	. . . . . . 7 ⊢ -∞ < +∞
20		breq1 4033	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
21	19, 20	mpbiri 168	. . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞)
22	21	necon3bi 2414	. . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞)
23	22	necon2bi 2419	. . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
24	18, 23	2falsed 703	. . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
25	7, 14, 24	3jaoi 1314	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
26	1, 25	sylbi 121	1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 105   ∨ w3o 979   = wceq 1364   ∈ wcel 2164   class class class wbr 4030  ℝcr 7873  +∞cpnf 8053  -∞cmnf 8054  ℝ^*cxr 8055   < clt 8056

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 615  ax-in2 616  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-13 2166  ax-14 2167  ax-ext 2175  ax-sep 4148  ax-pow 4204  ax-pr 4239  ax-un 4465  ax-setind 4570  ax-cnex 7965  ax-resscn 7966  ax-pre-ltirr 7986

This theorem depends on definitions:  df-bi 117  df-3or 981  df-3an 982  df-tru 1367  df-fal 1370  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ne 2365  df-nel 2460  df-ral 2477  df-rex 2478  df-rab 2481  df-v 2762  df-dif 3156  df-un 3158  df-in 3160  df-ss 3167  df-pw 3604  df-sn 3625  df-pr 3626  df-op 3628  df-uni 3837  df-br 4031  df-opab 4092  df-xp 4666  df-pnf 8058  df-mnf 8059  df-xr 8060  df-ltxr 8061

This theorem is referenced by:  npnflt  9884  xgepnf  9885  xrmaxiflemlub  11394