Theorem xrlttri 12573

Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 10649 or axlttri 10750. (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 12555	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
2	1	adantr 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3		breq2 5036	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
4	3	adantl 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
5	2, 4	mtbid 327	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
6	5	ex 416	. . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
7	6	adantr 484	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8		xrltnsym 12571	. . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
9	8	ancoms 462	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
10	7, 9	jaod 856	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11		elxr 12552	. . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12		elxr 12552	. . . 4 ⊢ (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13		axlttri 10750	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
14	13	biimprd 251	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → 𝐴 < 𝐵))
15	14	con1d 147	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
16		ltpnf 12556	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
17	16	adantr 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18		breq2 5036	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
19	18	adantl 485	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
20	17, 19	mpbird 260	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
21	20	pm2.24d 154	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
22		mnflt 12559	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
23	22	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24		breq1 5035	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
25	24	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
26	23, 25	mpbird 260	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
27	26	olcd 871	. . . . . . 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 1427	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
30		ltpnf 12556	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
31	30	adantl 485	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32		breq2 5036	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
33	32	adantr 484	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
34	31, 33	mpbird 260	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
35	34	olcd 871	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
36	35	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
37		eqtr3 2780	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
38	37	orcd 870	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
39	38	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
40		mnfltpnf 12562	. . . . . . . . . 10 ⊢ -∞ < +∞
41		breq12 5037	. . . . . . . . . 10 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
42	40, 41	mpbiri 261	. . . . . . . . 9 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
43	42	ancoms 462	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
44	43	olcd 871	. . . . . . 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 1427	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
47		mnflt 12559	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
48	47	adantl 485	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49		breq1 5035	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
50	49	adantr 484	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
51	48, 50	mpbird 260	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
52	51	pm2.24d 154	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
53		breq12 5037	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
54	40, 53	mpbiri 261	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
55	54	pm2.24d 154	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
56		eqtr3 2780	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
57	56	orcd 870	. . . . . . 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 1427	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
60	29, 46, 59	3jaoian 1426	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
61	11, 12, 60	syl2anb 600	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
62	10, 61	impbid 215	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
63	62	con2bid 358	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 209   ∧ wa 399   ∨ wo 844   ∨ w3o 1083   = wceq 1538   ∈ wcel 2111   class class class wbr 5032  ℝcr 10574  +∞cpnf 10710  -∞cmnf 10711  ℝ^*cxr 10712   < clt 10713

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-10 2142  ax-11 2158  ax-12 2175  ax-ext 2729  ax-sep 5169  ax-nul 5176  ax-pow 5234  ax-pr 5298  ax-un 7459  ax-cnex 10631  ax-resscn 10632  ax-pre-lttri 10649  ax-pre-lttrn 10650

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-3or 1085  df-3an 1086  df-tru 1541  df-fal 1551  df-ex 1782  df-nf 1786  df-sb 2070  df-mo 2557  df-eu 2588  df-clab 2736  df-cleq 2750  df-clel 2830  df-nfc 2901  df-ne 2952  df-nel 3056  df-ral 3075  df-rex 3076  df-rab 3079  df-v 3411  df-sbc 3697  df-csb 3806  df-dif 3861  df-un 3863  df-in 3865  df-ss 3875  df-nul 4226  df-if 4421  df-pw 4496  df-sn 4523  df-pr 4525  df-op 4529  df-uni 4799  df-br 5033  df-opab 5095  df-mpt 5113  df-id 5430  df-po 5443  df-so 5444  df-xp 5530  df-rel 5531  df-cnv 5532  df-co 5533  df-dm 5534  df-rn 5535  df-res 5536  df-ima 5537  df-iota 6294  df-fun 6337  df-fn 6338  df-f 6339  df-f1 6340  df-fo 6341  df-f1o 6342  df-fv 6343  df-er 8299  df-en 8528  df-dom 8529  df-sdom 8530  df-pnf 10715  df-mnf 10716  df-xr 10717  df-ltxr 10718

This theorem is referenced by:  xrltso  12575  xrleloe  12578  xrltlen  12580