Theorem xrlttri 12533

Description: Ordering on the extended reals satisfies strict trichotomy. New proofs should generally use this instead of ax-pre-lttri 10611 or axlttri 10712. (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 12515	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ¬ 𝐴 < 𝐴)
2	1	adantr 483	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐴)
3		breq2 5070	. . . . . . . 8 ⊢ (𝐴 = 𝐵 → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
4	3	adantl 484	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → (𝐴 < 𝐴 ↔ 𝐴 < 𝐵))
5	2, 4	mtbid 326	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐴 = 𝐵) → ¬ 𝐴 < 𝐵)
6	5	ex 415	. . . . 5 ⊢ (𝐴 ∈ ℝ* → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
7	6	adantr 483	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 = 𝐵 → ¬ 𝐴 < 𝐵))
8		xrltnsym 12531	. . . . 5 ⊢ ((𝐵 ∈ ℝ* ∧ 𝐴 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
9	8	ancoms 461	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐵 < 𝐴 → ¬ 𝐴 < 𝐵))
10	7, 9	jaod 855	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) → ¬ 𝐴 < 𝐵))
11		elxr 12512	. . . 4 ⊢ (𝐴 ∈ ℝ* ↔ (𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞))
12		elxr 12512	. . . 4 ⊢ (𝐵 ∈ ℝ* ↔ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞))
13		axlttri 10712	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
14	13	biimprd 250	. . . . . . 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 12516	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ → 𝐴 < +∞)
17	16	adantr 483	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < +∞)
18		breq2 5070	. . . . . . . . 9 ⊢ (𝐵 = +∞ → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
19	18	adantl 484	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ 𝐴 < +∞))
20	17, 19	mpbird 259	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
21	20	pm2.24d 154	. . . . . 6 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
22		mnflt 12519	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ → -∞ < 𝐴)
23	22	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → -∞ < 𝐴)
24		breq1 5069	. . . . . . . . . 10 ⊢ (𝐵 = -∞ → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
25	24	adantl 484	. . . . . . . . 9 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → (𝐵 < 𝐴 ↔ -∞ < 𝐴))
26	23, 25	mpbird 259	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
27	26	olcd 870	. . . . . . 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 1426	. . . . 5 ⊢ ((𝐴 ∈ ℝ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
30		ltpnf 12516	. . . . . . . . . 10 ⊢ (𝐵 ∈ ℝ → 𝐵 < +∞)
31	30	adantl 484	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < +∞)
32		breq2 5070	. . . . . . . . . 10 ⊢ (𝐴 = +∞ → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
33	32	adantr 483	. . . . . . . . 9 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐵 < 𝐴 ↔ 𝐵 < +∞))
34	31, 33	mpbird 259	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → 𝐵 < 𝐴)
35	34	olcd 870	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
36	35	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
37		eqtr3 2843	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → 𝐴 = 𝐵)
38	37	orcd 869	. . . . . . 7 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴))
39	38	a1d 25	. . . . . 6 ⊢ ((𝐴 = +∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
40		mnfltpnf 12522	. . . . . . . . . 10 ⊢ -∞ < +∞
41		breq12 5071	. . . . . . . . . 10 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → (𝐵 < 𝐴 ↔ -∞ < +∞))
42	40, 41	mpbiri 260	. . . . . . . . 9 ⊢ ((𝐵 = -∞ ∧ 𝐴 = +∞) → 𝐵 < 𝐴)
43	42	ancoms 461	. . . . . . . 8 ⊢ ((𝐴 = +∞ ∧ 𝐵 = -∞) → 𝐵 < 𝐴)
44	43	olcd 870	. . . . . . 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 1426	. . . . 5 ⊢ ((𝐴 = +∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
47		mnflt 12519	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ → -∞ < 𝐵)
48	47	adantl 484	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → -∞ < 𝐵)
49		breq1 5069	. . . . . . . . 9 ⊢ (𝐴 = -∞ → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
50	49	adantr 483	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ -∞ < 𝐵))
51	48, 50	mpbird 259	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → 𝐴 < 𝐵)
52	51	pm2.24d 154	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 ∈ ℝ) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
53		breq12 5071	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (𝐴 < 𝐵 ↔ -∞ < +∞))
54	40, 53	mpbiri 260	. . . . . . 7 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → 𝐴 < 𝐵)
55	54	pm2.24d 154	. . . . . 6 ⊢ ((𝐴 = -∞ ∧ 𝐵 = +∞) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
56		eqtr3 2843	. . . . . . . 8 ⊢ ((𝐴 = -∞ ∧ 𝐵 = -∞) → 𝐴 = 𝐵)
57	56	orcd 869	. . . . . . 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 1426	. . . . 5 ⊢ ((𝐴 = -∞ ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
60	29, 46, 59	3jaoian 1425	. . . 4 ⊢ (((𝐴 ∈ ℝ ∨ 𝐴 = +∞ ∨ 𝐴 = -∞) ∧ (𝐵 ∈ ℝ ∨ 𝐵 = +∞ ∨ 𝐵 = -∞)) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
61	11, 12, 60	syl2anb 599	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (¬ 𝐴 < 𝐵 → (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))
62	10, 61	impbid 214	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 = 𝐵 ∨ 𝐵 < 𝐴) ↔ ¬ 𝐴 < 𝐵))
63	62	con2bid 357	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ¬ (𝐴 = 𝐵 ∨ 𝐵 < 𝐴)))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 208   ∧ wa 398   ∨ wo 843   ∨ w3o 1082   = wceq 1537   ∈ wcel 2114   class class class wbr 5066  ℝcr 10536  +∞cpnf 10672  -∞cmnf 10673  ℝ^*cxr 10674   < clt 10675

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2793  ax-sep 5203  ax-nul 5210  ax-pow 5266  ax-pr 5330  ax-un 7461  ax-cnex 10593  ax-resscn 10594  ax-pre-lttri 10611  ax-pre-lttrn 10612

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-3or 1084  df-3an 1085  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-mo 2622  df-eu 2654  df-clab 2800  df-cleq 2814  df-clel 2893  df-nfc 2963  df-ne 3017  df-nel 3124  df-ral 3143  df-rex 3144  df-rab 3147  df-v 3496  df-sbc 3773  df-csb 3884  df-dif 3939  df-un 3941  df-in 3943  df-ss 3952  df-nul 4292  df-if 4468  df-pw 4541  df-sn 4568  df-pr 4570  df-op 4574  df-uni 4839  df-br 5067  df-opab 5129  df-mpt 5147  df-id 5460  df-po 5474  df-so 5475  df-xp 5561  df-rel 5562  df-cnv 5563  df-co 5564  df-dm 5565  df-rn 5566  df-res 5567  df-ima 5568  df-iota 6314  df-fun 6357  df-fn 6358  df-f 6359  df-f1 6360  df-fo 6361  df-f1o 6362  df-fv 6363  df-er 8289  df-en 8510  df-dom 8511  df-sdom 8512  df-pnf 10677  df-mnf 10678  df-xr 10679  df-ltxr 10680

This theorem is referenced by:  xrltso  12535  xrleloe  12538  xrltlen  12540