Theorem nltpnft 10018

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 9980	. 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8202	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2421	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4		ltpnf 9984	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
5		notnot 632	. . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
6	4, 5	syl 14	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
7	3, 6	2falsed 707	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8		id 19	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
9		pnfxr 8207	. . . . . 6 ⊢ +∞ ∈ ℝ*
10		xrltnr 9983	. . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ¬ +∞ < +∞
12		breq1 4086	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
13	11, 12	mtbiri 679	. . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
14	8, 13	2thd 175	. . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15		mnfnepnf 8210	. . . . . 6 ⊢ -∞ ≠ +∞
16	15	neii 2402	. . . . 5 ⊢ ¬ -∞ = +∞
17		eqeq1 2236	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
18	16, 17	mtbiri 679	. . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
19		mnfltpnf 9989	. . . . . . 7 ⊢ -∞ < +∞
20		breq1 4086	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
21	19, 20	mpbiri 168	. . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞)
22	21	necon3bi 2450	. . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞)
23	22	necon2bi 2455	. . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
24	18, 23	2falsed 707	. . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
25	7, 14, 24	3jaoi 1337	. 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 1001   = wceq 1395   ∈ wcel 2200   class class class wbr 4083  ℝcr 8006  +∞cpnf 8186  -∞cmnf 8187  ℝ^*cxr 8188   < clt 8189

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 617  ax-in2 618  ax-io 714  ax-5 1493  ax-7 1494  ax-gen 1495  ax-ie1 1539  ax-ie2 1540  ax-8 1550  ax-10 1551  ax-11 1552  ax-i12 1553  ax-bndl 1555  ax-4 1556  ax-17 1572  ax-i9 1576  ax-ial 1580  ax-i5r 1581  ax-13 2202  ax-14 2203  ax-ext 2211  ax-sep 4202  ax-pow 4258  ax-pr 4293  ax-un 4524  ax-setind 4629  ax-cnex 8098  ax-resscn 8099  ax-pre-ltirr 8119

This theorem depends on definitions:  df-bi 117  df-3or 1003  df-3an 1004  df-tru 1398  df-fal 1401  df-nf 1507  df-sb 1809  df-eu 2080  df-mo 2081  df-clab 2216  df-cleq 2222  df-clel 2225  df-nfc 2361  df-ne 2401  df-nel 2496  df-ral 2513  df-rex 2514  df-rab 2517  df-v 2801  df-dif 3199  df-un 3201  df-in 3203  df-ss 3210  df-pw 3651  df-sn 3672  df-pr 3673  df-op 3675  df-uni 3889  df-br 4084  df-opab 4146  df-xp 4725  df-pnf 8191  df-mnf 8192  df-xr 8193  df-ltxr 8194

This theorem is referenced by:  npnflt  10019  xgepnf  10020  xrmaxiflemlub  11767