Theorem ltxr 13131

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 5110	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
2		df-3an 1103	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 5172	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4	1, 3	brab2a 5745	. . . 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 5156	. . . 4 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
7		brxp 5701	. . . . . . 7 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
8		elun 4109	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}))
9		orcom 883	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
10	8, 9	bitri 278	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
11		elsng 4599	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ {-∞} ↔ 𝐴 = -∞))
12	11	orbi1d 929	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
13	10, 12	bitrid 286	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
14		elsng 4599	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ {+∞} ↔ 𝐵 = +∞))
15	13, 14	bi2anan9 649	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞)))
16		andir 1024	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)))
17	15, 16	bitrdi 290	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
18	7, 17	bitrid 286	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
19		brxp 5701	. . . . . . 7 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
20	11	anbi1d 642	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
21	20	adantr 485	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
22	19, 21	bitrid 286	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
23	18, 22	orbi12d 931	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ (((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
24		orass 934	. . . . 5 ⊢ ((((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
25	23, 24	bitrdi 290	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
26	6, 25	bitrid 286	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
27	5, 26	orbi12d 931	. 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 11236	. . . 4 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
29	28	breqi 5111	. . 3 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
30		brun 5156	. . 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 278	. 2 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
32		orass 934	. 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 317	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 209   ∧ wa 400   ∨ wo 860   ∧ w3a 1101   = wceq 1563   ∈ wcel 2145   ∪ cun 3905  {csn 4585   class class class wbr 5105  {copab 5167   × cxp 5650  ℝcr 11087   <_ℝ cltrr 11092  +∞cpnf 11228  -∞cmnf 11229  ℝ^*cxr 11230   < clt 11231

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737  ax-sep 5251  ax-pr 5395

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rex 3090  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-in 3914  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-br 5106  df-opab 5168  df-xp 5658  df-ltxr 11236

This theorem is referenced by:  xrltnr  13135  ltpnf  13136  mnflt  13139  mnfltpnf  13142  pnfnlt  13144  nltmnf  13145