Theorem ltxr 12957

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 5102	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
2		df-3an 1089	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 5164	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4	1, 3	brab2a 5716	. . . 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 5148	. . . 4 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
7		brxp 5672	. . . . . . 7 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
8		elun 4100	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}))
9		orcom 868	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
10	8, 9	bitri 275	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
11		elsng 4592	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ {-∞} ↔ 𝐴 = -∞))
12	11	orbi1d 915	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
13	10, 12	bitrid 283	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
14		elsng 4592	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ {+∞} ↔ 𝐵 = +∞))
15	13, 14	bi2anan9 637	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞)))
16		andir 1007	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)))
17	15, 16	bitrdi 287	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
18	7, 17	bitrid 283	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
19		brxp 5672	. . . . . . 7 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
20	11	anbi1d 631	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
21	20	adantr 482	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
22	19, 21	bitrid 283	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
23	18, 22	orbi12d 917	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ (((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
24		orass 920	. . . . 5 ⊢ ((((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
25	23, 24	bitrdi 287	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
26	6, 25	bitrid 283	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
27	5, 26	orbi12d 917	. 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 11120	. . . 4 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
29	28	breqi 5103	. . 3 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
30		brun 5148	. . 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 275	. 2 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
32		orass 920	. 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 314	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 205   ∧ wa 397   ∨ wo 845   ∧ w3a 1087   = wceq 1541   ∈ wcel 2106   ∪ cun 3900  {csn 4578   class class class wbr 5097  {copab 5159   × cxp 5623  ℝcr 10976   <_ℝ cltrr 10981  +∞cpnf 11112  -∞cmnf 11113  ℝ^*cxr 11114   < clt 11115

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708  ax-sep 5248  ax-nul 5255  ax-pr 5377

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-3an 1089  df-tru 1544  df-fal 1554  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-ral 3063  df-rex 3072  df-rab 3405  df-v 3444  df-dif 3905  df-un 3907  df-in 3909  df-ss 3919  df-nul 4275  df-if 4479  df-sn 4579  df-pr 4581  df-op 4585  df-br 5098  df-opab 5160  df-xp 5631  df-ltxr 11120

This theorem is referenced by:  xrltnr  12961  ltpnf  12962  mnflt  12965  mnfltpnf  12968  pnfnlt  12970  nltmnf  12971