Theorem nltpnft 9978

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 9940	. 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8162	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2401	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4		ltpnf 9944	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
5		notnot 632	. . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
6	4, 5	syl 14	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
7	3, 6	2falsed 706	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8		id 19	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
9		pnfxr 8167	. . . . . 6 ⊢ +∞ ∈ ℝ*
10		xrltnr 9943	. . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ¬ +∞ < +∞
12		breq1 4065	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
13	11, 12	mtbiri 679	. . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
14	8, 13	2thd 175	. . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15		mnfnepnf 8170	. . . . . 6 ⊢ -∞ ≠ +∞
16	15	neii 2382	. . . . 5 ⊢ ¬ -∞ = +∞
17		eqeq1 2216	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
18	16, 17	mtbiri 679	. . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
19		mnfltpnf 9949	. . . . . . 7 ⊢ -∞ < +∞
20		breq1 4065	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
21	19, 20	mpbiri 168	. . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞)
22	21	necon3bi 2430	. . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞)
23	22	necon2bi 2435	. . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
24	18, 23	2falsed 706	. . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
25	7, 14, 24	3jaoi 1318	. 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 982   = wceq 1375   ∈ wcel 2180   class class class wbr 4062  ℝcr 7966  +∞cpnf 8146  -∞cmnf 8147  ℝ^*cxr 8148   < clt 8149

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 713  ax-5 1473  ax-7 1474  ax-gen 1475  ax-ie1 1519  ax-ie2 1520  ax-8 1530  ax-10 1531  ax-11 1532  ax-i12 1533  ax-bndl 1535  ax-4 1536  ax-17 1552  ax-i9 1556  ax-ial 1560  ax-i5r 1561  ax-13 2182  ax-14 2183  ax-ext 2191  ax-sep 4181  ax-pow 4237  ax-pr 4272  ax-un 4501  ax-setind 4606  ax-cnex 8058  ax-resscn 8059  ax-pre-ltirr 8079

This theorem depends on definitions:  df-bi 117  df-3or 984  df-3an 985  df-tru 1378  df-fal 1381  df-nf 1487  df-sb 1789  df-eu 2060  df-mo 2061  df-clab 2196  df-cleq 2202  df-clel 2205  df-nfc 2341  df-ne 2381  df-nel 2476  df-ral 2493  df-rex 2494  df-rab 2497  df-v 2781  df-dif 3179  df-un 3181  df-in 3183  df-ss 3190  df-pw 3631  df-sn 3652  df-pr 3653  df-op 3655  df-uni 3868  df-br 4063  df-opab 4125  df-xp 4702  df-pnf 8151  df-mnf 8152  df-xr 8153  df-ltxr 8154

This theorem is referenced by:  npnflt  9979  xgepnf  9980  xrmaxiflemlub  11725