Theorem nltpnft 8730

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 8696	. 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 7073	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2226	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4		ltpnf 8702	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
5		notnot 559	. . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
6	4, 5	syl 14	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
7	3, 6	2falsed 618	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8		id 19	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
9		pnfxr 8692	. . . . . 6 ⊢ +∞ ∈ ℝ*
10		xrltnr 8701	. . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
11	9, 10	ax-mp 7	. . . . 5 ⊢ ¬ +∞ < +∞
12		breq1 3767	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
13	11, 12	mtbiri 600	. . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
14	8, 13	2thd 164	. . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15		mnfnepnf 8698	. . . . . 6 ⊢ -∞ ≠ +∞
16	15	neii 2208	. . . . 5 ⊢ ¬ -∞ = +∞
17		eqeq1 2046	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
18	16, 17	mtbiri 600	. . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
19		mnfltpnf 8706	. . . . . . 7 ⊢ -∞ < +∞
20		breq1 3767	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
21	19, 20	mpbiri 157	. . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞)
22	21	necon3bi 2255	. . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞)
23	22	necon2bi 2260	. . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
24	18, 23	2falsed 618	. . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
25	7, 14, 24	3jaoi 1198	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
26	1, 25	sylbi 114	1 ⊢ (𝐴 ∈ ℝ* → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 98   ∨ w3o 884   = wceq 1243   ∈ wcel 1393   class class class wbr 3764  ℝcr 6888  +∞cpnf 7057  -∞cmnf 7058  ℝ^*cxr 7059   < clt 7060

This theorem was proved from axioms:  ax-1 5  ax-2 6  ax-mp 7  ax-ia1 99  ax-ia2 100  ax-ia3 101  ax-in1 544  ax-in2 545  ax-io 630  ax-5 1336  ax-7 1337  ax-gen 1338  ax-ie1 1382  ax-ie2 1383  ax-8 1395  ax-10 1396  ax-11 1397  ax-i12 1398  ax-bndl 1399  ax-4 1400  ax-13 1404  ax-14 1405  ax-17 1419  ax-i9 1423  ax-ial 1427  ax-i5r 1428  ax-ext 2022  ax-sep 3875  ax-pow 3927  ax-pr 3944  ax-un 4170  ax-setind 4262  ax-cnex 6975  ax-resscn 6976  ax-pre-ltirr 6996

This theorem depends on definitions:  df-bi 110  df-3or 886  df-3an 887  df-tru 1246  df-fal 1249  df-nf 1350  df-sb 1646  df-eu 1903  df-mo 1904  df-clab 2027  df-cleq 2033  df-clel 2036  df-nfc 2167  df-ne 2206  df-nel 2207  df-ral 2311  df-rex 2312  df-rab 2315  df-v 2559  df-dif 2920  df-un 2922  df-in 2924  df-ss 2931  df-pw 3361  df-sn 3381  df-pr 3382  df-op 3384  df-uni 3581  df-br 3765  df-opab 3819  df-xp 4351  df-pnf 7062  df-mnf 7063  df-xr 7064  df-ltxr 7065

This theorem is referenced by: (None)