Theorem nltpnft 10153

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 10115	. 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		renepnf 8326	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 ≠ +∞)
3	2	neneqd 2435	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 = +∞)
4		ltpnf 10119	. . . . 5 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
5		notnot 634	. . . . 5 ⊢ (𝐴 < +∞ → ¬ ¬ 𝐴 < +∞)
6	4, 5	syl 14	. . . 4 ⊢ (𝐴 ∈ ℝ → ¬ ¬ 𝐴 < +∞)
7	3, 6	2falsed 710	. . 3 ⊢ (𝐴 ∈ ℝ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
8		id 19	. . . 4 ⊢ (𝐴 = +∞ → 𝐴 = +∞)
9		pnfxr 8331	. . . . . 6 ⊢ +∞ ∈ ℝ*
10		xrltnr 10118	. . . . . 6 ⊢ (+∞ ∈ ℝ* → ¬ +∞ < +∞)
11	9, 10	ax-mp 5	. . . . 5 ⊢ ¬ +∞ < +∞
12		breq1 4114	. . . . 5 ⊢ (𝐴 = +∞ → (𝐴 < +∞ ↔ +∞ < +∞))
13	11, 12	mtbiri 682	. . . 4 ⊢ (𝐴 = +∞ → ¬ 𝐴 < +∞)
14	8, 13	2thd 175	. . 3 ⊢ (𝐴 = +∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
15		mnfnepnf 8334	. . . . . 6 ⊢ -∞ ≠ +∞
16	15	neii 2416	. . . . 5 ⊢ ¬ -∞ = +∞
17		eqeq1 2241	. . . . 5 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ -∞ = +∞))
18	16, 17	mtbiri 682	. . . 4 ⊢ (𝐴 = -∞ → ¬ 𝐴 = +∞)
19		mnfltpnf 10124	. . . . . . 7 ⊢ -∞ < +∞
20		breq1 4114	. . . . . . 7 ⊢ (𝐴 = -∞ → (𝐴 < +∞ ↔ -∞ < +∞))
21	19, 20	mpbiri 168	. . . . . 6 ⊢ (𝐴 = -∞ → 𝐴 < +∞)
22	21	necon3bi 2464	. . . . 5 ⊢ (¬ 𝐴 < +∞ → 𝐴 ≠ -∞)
23	22	necon2bi 2469	. . . 4 ⊢ (𝐴 = -∞ → ¬ ¬ 𝐴 < +∞)
24	18, 23	2falsed 710	. . 3 ⊢ (𝐴 = -∞ → (𝐴 = +∞ ↔ ¬ 𝐴 < +∞))
25	7, 14, 24	3jaoi 1340	. 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 1004   = wceq 1398   ∈ wcel 2205   class class class wbr 4111  ℝcr 8131  +∞cpnf 8310  -∞cmnf 8311  ℝ^*cxr 8312   < clt 8313

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 619  ax-in2 620  ax-io 717  ax-5 1496  ax-7 1497  ax-gen 1498  ax-ie1 1542  ax-ie2 1543  ax-8 1553  ax-10 1554  ax-11 1555  ax-i12 1556  ax-bndl 1558  ax-4 1559  ax-17 1575  ax-i9 1579  ax-ial 1583  ax-i5r 1584  ax-13 2207  ax-14 2208  ax-ext 2216  ax-sep 4230  ax-pow 4289  ax-pr 4324  ax-un 4556  ax-setind 4661  ax-cnex 8223  ax-resscn 8224  ax-pre-ltirr 8244

This theorem depends on definitions:  df-bi 117  df-3or 1006  df-3an 1007  df-tru 1401  df-fal 1404  df-nf 1510  df-sb 1812  df-eu 2085  df-mo 2086  df-clab 2221  df-cleq 2227  df-clel 2230  df-nfc 2375  df-ne 2415  df-nel 2510  df-ral 2527  df-rex 2528  df-rab 2531  df-v 2817  df-dif 3215  df-un 3217  df-in 3219  df-ss 3226  df-pw 3673  df-sn 3697  df-pr 3698  df-op 3700  df-uni 3917  df-br 4112  df-opab 4174  df-xp 4757  df-pnf 8315  df-mnf 8316  df-xr 8317  df-ltxr 8318

This theorem is referenced by:  npnflt  10154  xgepnf  10155  xrmaxiflemlub  11941