Theorem xrlttri 13081

Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 11103 or axlttri 11208. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
xrlttri	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))

Proof of Theorem xrlttri

Step	Hyp	Ref	Expression
1		xrltnr 13061	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
2	1	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3		breq2 5076	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
4	3	adantl 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
5	2, 4	mtbid 325	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
6	5	ex 413	. . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
7	6	adantr 481	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8		xrltnsym 13079	. . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
9	8	ancoms 459	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
10	7, 9	jaod 865	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11		elxr 13058	. . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12		elxr 13058	. . . 4 ⊢ (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13		axlttri 11208	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
14	13	biimprd 249	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → 𝐴 < 𝐵))
15	14	con1d 145	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
16		ltpnf 13062	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
17	16	adantr 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18		breq2 5076	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
19	18	adantl 482	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
20	17, 19	mpbird 258	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
21	20	pm2.24d 151	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
22		mnflt 13065	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
23	22	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24		breq1 5075	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
25	24	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
26	23, 25	mpbird 258	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
27	26	olcd 880	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
28	27	a1d 25	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
29	15, 21, 28	3jaodan 1439	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
30		ltpnf 13062	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
31	30	adantl 482	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32		breq2 5076	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
33	32	adantr 481	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
34	31, 33	mpbird 258	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
35	34	olcd 880	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
36	35	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
37		eqtr3 2761	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
38	37	orcd 879	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
39	38	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
40		mnfltpnf 13068	. . . . . . . . . 10 ⊢ -∞ < +∞
41		breq12 5077	. . . . . . . . . 10 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
42	40, 41	mpbiri 259	. . . . . . . . 9 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
43	42	ancoms 459	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
44	43	olcd 880	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
45	44	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
46	36, 39, 45	3jaodan 1439	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
47		mnflt 13065	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
48	47	adantl 482	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49		breq1 5075	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
50	49	adantr 481	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
51	48, 50	mpbird 258	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
52	51	pm2.24d 151	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
53		breq12 5077	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
54	40, 53	mpbiri 259	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
55	54	pm2.24d 151	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
56		eqtr3 2761	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
57	56	orcd 879	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
58	57	a1d 25	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
59	52, 55, 58	3jaodan 1439	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
60	29, 46, 59	3jaoian 1438	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
61	11, 12, 60	syl2anb 604	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
62	10, 61	impbid 213	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
63	62	con2bid 355	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∨ w3o 1091   = wceq 1547   ∈ wcel 2119   class class class wbr 5072  ℝcr 11028  +∞cpnf 11167  -∞cmnf 11168  ℝ^*cxr 11169   < clt 11170

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1802  ax-4 1816  ax-5 1917  ax-6 1974  ax-7 2015  ax-8 2121  ax-9 2129  ax-10 2152  ax-11 2168  ax-12 2189  ax-ext 2711  ax-sep 5218  ax-nul 5228  ax-pow 5294  ax-pr 5362  ax-un 7678  ax-cnex 11085  ax-resscn 11086  ax-pre-lttri 11103  ax-pre-lttrn 11104

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3or 1093  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-nf 1791  df-sb 2074  df-mo 2543  df-eu 2573  df-clab 2718  df-cleq 2731  df-clel 2814  df-nfc 2888  df-ne 2935  df-nel 3039  df-ral 3054  df-rex 3064  df-rab 3392  df-v 3433  df-sbc 3724  df-csb 3832  df-dif 3886  df-un 3888  df-in 3890  df-ss 3900  df-nul 4262  df-if 4455  df-pw 4531  df-sn 4556  df-pr 4558  df-op 4562  df-uni 4839  df-br 5073  df-opab 5135  df-mpt 5154  df-id 5513  df-po 5526  df-so 5527  df-xp 5624  df-rel 5625  df-cnv 5626  df-co 5627  df-dm 5628  df-rn 5629  df-res 5630  df-ima 5631  df-iota 6441  df-fun 6487  df-fn 6488  df-f 6489  df-f1 6490  df-fo 6491  df-f1o 6492  df-fv 6493  df-er 8633  df-en 8884  df-dom 8885  df-sdom 8886  df-pnf 11172  df-mnf 11173  df-xr 11174  df-ltxr 11175

This theorem is referenced by:  xrltso  13083  xrleloe  13086  xrltlen  13088