Theorem xrltnr 9596

Description: The extended real 'less than' is irreflexive. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
xrltnr	⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)

Proof of Theorem xrltnr

Step	Hyp	Ref	Expression
1		elxr 9593	. 2 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
2		ltnr 7865	. . 3 ⊢ (𝐴 ∈ ℝ → ¬ 𝐴 < 𝐴)
3		pnfnre 7831	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
4	3	neli 2406	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
5	4	intnan 915	. . . . . . . 8 ⊢ ¬ (+∞ ∈ ℝ ∧ +∞ ∈ ℝ)
6	5	intnanr 916	. . . . . . 7 ⊢ ¬ ((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ <ℝ +∞)
7		pnfnemnf 7844	. . . . . . . . 9 ⊢ +∞ ≠ -∞
8	7	neii 2311	. . . . . . . 8 ⊢ ¬ +∞ = -∞
9	8	intnanr 916	. . . . . . 7 ⊢ ¬ (+∞ = -∞ ∧ +∞ = +∞)
10	6, 9	pm3.2ni 803	. . . . . 6 ⊢ ¬ (((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ <ℝ +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞))
11	4	intnanr 916	. . . . . . 7 ⊢ ¬ (+∞ ∈ ℝ ∧ +∞ = +∞)
12	4	intnan 915	. . . . . . 7 ⊢ ¬ (+∞ = -∞ ∧ +∞ ∈ ℝ)
13	11, 12	pm3.2ni 803	. . . . . 6 ⊢ ¬ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ))
14	10, 13	pm3.2ni 803	. . . . 5 ⊢ ¬ ((((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ <ℝ +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞)) ∨ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ)))
15		pnfxr 7842	. . . . . 6 ⊢ +∞ ∈ ℝ*
16		ltxr 9592	. . . . . 6 ⊢ ((+∞ ∈ ℝ* ∧ +∞ ∈ ℝ*) → (+∞ < +∞ ↔ ((((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ <ℝ +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞)) ∨ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ)))))
17	15, 15, 16	mp2an 423	. . . . 5 ⊢ (+∞ < +∞ ↔ ((((+∞ ∈ ℝ ∧ +∞ ∈ ℝ) ∧ +∞ <ℝ +∞) ∨ (+∞ = -∞ ∧ +∞ = +∞)) ∨ ((+∞ ∈ ℝ ∧ +∞ = +∞) ∨ (+∞ = -∞ ∧ +∞ ∈ ℝ))))
18	14, 17	mtbir 661	. . . 4 ⊢ ¬ +∞ < +∞
19		breq12 3942	. . . . 5 ⊢ ((𝐴 = +∞ ∧ 𝐴 = +∞) → (𝐴 < 𝐴 ↔ +∞ < +∞))
20	19	anidms 395	. . . 4 ⊢ (𝐴 = +∞ → (𝐴 < 𝐴 ↔ +∞ < +∞))
21	18, 20	mtbiri 665	. . 3 ⊢ (𝐴 = +∞ → ¬ 𝐴 < 𝐴)
22		mnfnre 7832	. . . . . . . . . 10 ⊢ -∞ ∉ ℝ
23	22	neli 2406	. . . . . . . . 9 ⊢ ¬ -∞ ∈ ℝ
24	23	intnan 915	. . . . . . . 8 ⊢ ¬ (-∞ ∈ ℝ ∧ -∞ ∈ ℝ)
25	24	intnanr 916	. . . . . . 7 ⊢ ¬ ((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ <ℝ -∞)
26	7	nesymi 2355	. . . . . . . 8 ⊢ ¬ -∞ = +∞
27	26	intnan 915	. . . . . . 7 ⊢ ¬ (-∞ = -∞ ∧ -∞ = +∞)
28	25, 27	pm3.2ni 803	. . . . . 6 ⊢ ¬ (((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ <ℝ -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞))
29	23	intnanr 916	. . . . . . 7 ⊢ ¬ (-∞ ∈ ℝ ∧ -∞ = +∞)
30	23	intnan 915	. . . . . . 7 ⊢ ¬ (-∞ = -∞ ∧ -∞ ∈ ℝ)
31	29, 30	pm3.2ni 803	. . . . . 6 ⊢ ¬ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ))
32	28, 31	pm3.2ni 803	. . . . 5 ⊢ ¬ ((((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ <ℝ -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ)))
33		mnfxr 7846	. . . . . 6 ⊢ -∞ ∈ ℝ*
34		ltxr 9592	. . . . . 6 ⊢ ((-∞ ∈ ℝ* ∧ -∞ ∈ ℝ*) → (-∞ < -∞ ↔ ((((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ <ℝ -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ)))))
35	33, 33, 34	mp2an 423	. . . . 5 ⊢ (-∞ < -∞ ↔ ((((-∞ ∈ ℝ ∧ -∞ ∈ ℝ) ∧ -∞ <ℝ -∞) ∨ (-∞ = -∞ ∧ -∞ = +∞)) ∨ ((-∞ ∈ ℝ ∧ -∞ = +∞) ∨ (-∞ = -∞ ∧ -∞ ∈ ℝ))))
36	32, 35	mtbir 661	. . . 4 ⊢ ¬ -∞ < -∞
37		breq12 3942	. . . . 5 ⊢ ((𝐴 = -∞ ∧ 𝐴 = -∞) → (𝐴 < 𝐴 ↔ -∞ < -∞))
38	37	anidms 395	. . . 4 ⊢ (𝐴 = -∞ → (𝐴 < 𝐴 ↔ -∞ < -∞))
39	36, 38	mtbiri 665	. . 3 ⊢ (𝐴 = -∞ → ¬ 𝐴 < 𝐴)
40	2, 21, 39	3jaoi 1282	. 2 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) → ¬ 𝐴 < 𝐴)
41	1, 40	sylbi 120	1 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 103   ↔ wb 104   ∨ wo 698   ∨ w3o 962   = wceq 1332   ∈ wcel 1481   class class class wbr 3937  ℝcr 7643   <_ℝ cltrr 7648  +∞cpnf 7821  -∞cmnf 7822  ℝ^*cxr 7823   < clt 7824

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-io 699  ax-5 1424  ax-7 1425  ax-gen 1426  ax-ie1 1470  ax-ie2 1471  ax-8 1483  ax-10 1484  ax-11 1485  ax-i12 1486  ax-bndl 1487  ax-4 1488  ax-13 1492  ax-14 1493  ax-17 1507  ax-i9 1511  ax-ial 1515  ax-i5r 1516  ax-ext 2122  ax-sep 4054  ax-pow 4106  ax-pr 4139  ax-un 4363  ax-setind 4460  ax-cnex 7735  ax-resscn 7736  ax-pre-ltirr 7756

This theorem depends on definitions:  df-bi 116  df-3or 964  df-3an 965  df-tru 1335  df-fal 1338  df-nf 1438  df-sb 1737  df-eu 2003  df-mo 2004  df-clab 2127  df-cleq 2133  df-clel 2136  df-nfc 2271  df-ne 2310  df-nel 2405  df-ral 2422  df-rex 2423  df-rab 2426  df-v 2691  df-dif 3078  df-un 3080  df-in 3082  df-ss 3089  df-pw 3517  df-sn 3538  df-pr 3539  df-op 3541  df-uni 3745  df-br 3938  df-opab 3998  df-xp 4553  df-pnf 7826  df-mnf 7827  df-xr 7828  df-ltxr 7829

This theorem is referenced by:  xrltnsym  9609  xrltso  9612  xrlttri3  9613  xrleid  9616  xrltne  9626  nltpnft  9627  ngtmnft  9630  xrrebnd  9632  xposdif  9695  lbioog  9726  ubioog  9727  xrmaxleim  11045  xrmaxiflemlub  11049