Theorem ltxr 13064

Description: The 'less than' binary relation on the set of extended reals. Definition 12-3.1 of [Gleason] p. 173. (Contributed by NM, 14-Oct-2005.)

Assertion

Ref	Expression
ltxr	⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))

Proof of Theorem ltxr

Dummy variables 𝑥 𝑦 are mutually distinct and distinct from all other variables.

Step	Hyp	Ref	Expression
1		breq12 5084	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
2		df-3an 1094	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 5146	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4	1, 3	brab2a 5718	. . . 4 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵))
5	4	a1i 11	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵)))
6		brun 5130	. . . 4 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
7		brxp 5674	. . . . . . 7 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
8		elun 4090	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}))
9		orcom 876	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
10	8, 9	bitri 276	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
11		elsng 4576	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ {-∞} ↔ 𝐴 = -∞))
12	11	orbi1d 922	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
13	10, 12	bitrid 284	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
14		elsng 4576	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ {+∞} ↔ 𝐵 = +∞))
15	13, 14	bi2anan9 644	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞)))
16		andir 1016	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)))
17	15, 16	bitrdi 288	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
18	7, 17	bitrid 284	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
19		brxp 5674	. . . . . . 7 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
20	11	anbi1d 637	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
21	20	adantr 481	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
22	19, 21	bitrid 284	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
23	18, 22	orbi12d 924	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ (((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
24		orass 927	. . . . 5 ⊢ ((((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
25	23, 24	bitrdi 288	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
26	6, 25	bitrid 284	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
27	5, 26	orbi12d 924	. 2 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵) ↔ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))))
28		df-ltxr 11182	. . . 4 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
29	28	breqi 5085	. . 3 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
30		brun 5130	. . 3 ⊢ (𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
31	29, 30	bitri 276	. 2 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
32		orass 927	. 2 ⊢ (((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))) ↔ (((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
33	27, 31, 32	3bitr4g 315	1 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴 < 𝐵 ↔ ((((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵) ∨ (𝐴 = -∞ ∧ 𝐵 = +∞)) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))

Syntax hints:   → wi 4   ↔ wb 207   ∧ wa 396   ∨ wo 853   ∧ w3a 1092   = wceq 1547   ∈ wcel 2119   ∪ cun 3888  {csn 4562   class class class wbr 5079  {copab 5141   × cxp 5623  ℝcr 11035   <_ℝ cltrr 11040  +∞cpnf 11174  -∞cmnf 11175  ℝ^*cxr 11176   < clt 11177

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-ext 2712  ax-sep 5225  ax-pr 5369

This theorem depends on definitions:  df-bi 208  df-an 397  df-or 854  df-3an 1094  df-tru 1550  df-fal 1560  df-ex 1787  df-sb 2074  df-clab 2719  df-cleq 2732  df-clel 2815  df-ral 3055  df-rex 3065  df-rab 3393  df-v 3434  df-dif 3893  df-un 3895  df-in 3897  df-ss 3907  df-nul 4269  df-if 4462  df-sn 4563  df-pr 4565  df-op 4569  df-br 5080  df-opab 5142  df-xp 5631  df-ltxr 11182

This theorem is referenced by:  xrltnr  13068  ltpnf  13069  mnflt  13072  mnfltpnf  13075  pnfnlt  13077  nltmnf  13078