Theorem xrlttri 13040

Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 11087 or axlttri 11191. (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 13020	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3		breq2 5097	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
4	3	adantl 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
5	2, 4	mtbid 324	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
6	5	ex 412	. . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
7	6	adantr 480	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8		xrltnsym 13038	. . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
9	8	ancoms 458	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
10	7, 9	jaod 859	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11		elxr 13017	. . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12		elxr 13017	. . . 4 ⊢ (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13		axlttri 11191	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
14	13	biimprd 248	. . . . . . 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 13021	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18		breq2 5097	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
19	18	adantl 481	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
20	17, 19	mpbird 257	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
21	20	pm2.24d 151	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
22		mnflt 13024	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24		breq1 5096	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
25	24	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
26	23, 25	mpbird 257	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
27	26	olcd 874	. . . . . . 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 1433	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
30		ltpnf 13021	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32		breq2 5097	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
33	32	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
34	31, 33	mpbird 257	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
35	34	olcd 874	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
36	35	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
37		eqtr3 2755	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
38	37	orcd 873	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
39	38	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
40		mnfltpnf 13027	. . . . . . . . . 10 ⊢ -∞ < +∞
41		breq12 5098	. . . . . . . . . 10 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
42	40, 41	mpbiri 258	. . . . . . . . 9 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
43	42	ancoms 458	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
44	43	olcd 874	. . . . . . 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 1433	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
47		mnflt 13024	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
48	47	adantl 481	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49		breq1 5096	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
50	49	adantr 480	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
51	48, 50	mpbird 257	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
52	51	pm2.24d 151	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
53		breq12 5098	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
54	40, 53	mpbiri 258	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
55	54	pm2.24d 151	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
56		eqtr3 2755	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
57	56	orcd 873	. . . . . . 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 1433	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
60	29, 46, 59	3jaoian 1432	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
61	11, 12, 60	syl2anb 598	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
62	10, 61	impbid 212	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
63	62	con2bid 354	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∨ w3o 1085   = wceq 1541   ∈ wcel 2113   class class class wbr 5093  ℝcr 11012  +∞cpnf 11150  -∞cmnf 11151  ℝ^*cxr 11152   < clt 11153

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2115  ax-9 2123  ax-10 2146  ax-11 2162  ax-12 2182  ax-ext 2705  ax-sep 5236  ax-nul 5246  ax-pow 5305  ax-pr 5372  ax-un 7674  ax-cnex 11069  ax-resscn 11070  ax-pre-lttri 11087  ax-pre-lttrn 11088

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2537  df-eu 2566  df-clab 2712  df-cleq 2725  df-clel 2808  df-nfc 2882  df-ne 2930  df-nel 3034  df-ral 3049  df-rex 3058  df-rab 3397  df-v 3439  df-sbc 3738  df-csb 3847  df-dif 3901  df-un 3903  df-in 3905  df-ss 3915  df-nul 4283  df-if 4475  df-pw 4551  df-sn 4576  df-pr 4578  df-op 4582  df-uni 4859  df-br 5094  df-opab 5156  df-mpt 5175  df-id 5514  df-po 5527  df-so 5528  df-xp 5625  df-rel 5626  df-cnv 5627  df-co 5628  df-dm 5629  df-rn 5630  df-res 5631  df-ima 5632  df-iota 6442  df-fun 6488  df-fn 6489  df-f 6490  df-f1 6491  df-fo 6492  df-f1o 6493  df-fv 6494  df-er 8628  df-en 8876  df-dom 8877  df-sdom 8878  df-pnf 11155  df-mnf 11156  df-xr 11157  df-ltxr 11158

This theorem is referenced by:  xrltso  13042  xrleloe  13045  xrltlen  13047