Theorem xrlttri 13075

Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 11118 or axlttri 11221. (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 13055	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
2	1	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3		breq2 5106	. . . . . . . 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 13073	. . . . 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 13052	. . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12		elxr 13052	. . . 4 ⊢ (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13		axlttri 11221	. . . . . . . 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 13056	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
17	16	adantr 480	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18		breq2 5106	. . . . . . . . 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 13059	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
23	22	adantr 480	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24		breq1 5105	. . . . . . . . . 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 13056	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
31	30	adantl 481	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32		breq2 5106	. . . . . . . . . 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 2751	. . . . . . . 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 13062	. . . . . . . . . 10 ⊢ -∞ < +∞
41		breq12 5107	. . . . . . . . . 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 13059	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
48	47	adantl 481	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49		breq1 5105	. . . . . . . . 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 5107	. . . . . . . 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 2751	. . . . . . . 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 1540   ∈ wcel 2109   class class class wbr 5102  ℝcr 11043  +∞cpnf 11181  -∞cmnf 11182  ℝ^*cxr 11183   < clt 11184

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-10 2142  ax-11 2158  ax-12 2178  ax-ext 2701  ax-sep 5246  ax-nul 5256  ax-pow 5315  ax-pr 5382  ax-un 7691  ax-cnex 11100  ax-resscn 11101  ax-pre-lttri 11118  ax-pre-lttrn 11119

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3or 1087  df-3an 1088  df-tru 1543  df-fal 1553  df-ex 1780  df-nf 1784  df-sb 2066  df-mo 2533  df-eu 2562  df-clab 2708  df-cleq 2721  df-clel 2803  df-nfc 2878  df-ne 2926  df-nel 3030  df-ral 3045  df-rex 3054  df-rab 3403  df-v 3446  df-sbc 3751  df-csb 3860  df-dif 3914  df-un 3916  df-in 3918  df-ss 3928  df-nul 4293  df-if 4485  df-pw 4561  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5103  df-opab 5165  df-mpt 5184  df-id 5526  df-po 5539  df-so 5540  df-xp 5637  df-rel 5638  df-cnv 5639  df-co 5640  df-dm 5641  df-rn 5642  df-res 5643  df-ima 5644  df-iota 6452  df-fun 6501  df-fn 6502  df-f 6503  df-f1 6504  df-fo 6505  df-f1o 6506  df-fv 6507  df-er 8648  df-en 8896  df-dom 8897  df-sdom 8898  df-pnf 11186  df-mnf 11187  df-xr 11188  df-ltxr 11189

This theorem is referenced by:  xrltso  13077  xrleloe  13080  xrltlen  13082