Theorem ltxr 9841

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 4034	. . . . 5 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
2		df-3an 982	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 4096	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4	1, 3	brab2ga 4734	. . . 4 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵))
5	4	a1i 9	. . 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 4080	. . . 4 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
7		brxp 4690	. . . . . . 7 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
8		elun 3300	. . . . . . . . . . 11 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}))
9		orcom 729	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∨ 𝐴 ∈ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
10	8, 9	bitri 184	. . . . . . . . . 10 ⊢ (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ))
11		elsng 3633	. . . . . . . . . . 11 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ {-∞} ↔ 𝐴 = -∞))
12	11	orbi1d 792	. . . . . . . . . 10 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∨ 𝐴 ∈ ℝ) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
13	10, 12	bitrid 192	. . . . . . . . 9 ⊢ (𝐴 ∈ ℝ* → (𝐴 ∈ (ℝ ∪ {-∞}) ↔ (𝐴 = -∞ ∨ 𝐴 ∈ ℝ)))
14		elsng 3633	. . . . . . . . 9 ⊢ (𝐵 ∈ ℝ* → (𝐵 ∈ {+∞} ↔ 𝐵 = +∞))
15	13, 14	bi2anan9 606	. . . . . . . 8 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞)))
16		andir 820	. . . . . . . 8 ⊢ (((𝐴 = -∞ ∨ 𝐴 ∈ ℝ) ∧ 𝐵 = +∞) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)))
17	15, 16	bitrdi 196	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
18	7, 17	bitrid 192	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞))))
19		brxp 4690	. . . . . . 7 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
20	11	anbi1d 465	. . . . . . . 8 ⊢ (𝐴 ∈ ℝ* → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
21	20	adantr 276	. . . . . . 7 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
22	19, 21	bitrid 192	. . . . . 6 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))
23	18, 22	orbi12d 794	. . . . 5 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ (((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
24		orass 768	. . . . 5 ⊢ ((((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ (𝐴 ∈ ℝ ∧ 𝐵 = +∞)) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ))))
25	23, 24	bitrdi 196	. . . 4 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
26	6, 25	bitrid 192	. . 3 ⊢ ((𝐴 ∈ ℝ* ∧ 𝐵 ∈ ℝ*) → (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ ((𝐴 = -∞ ∧ 𝐵 = +∞) ∨ ((𝐴 ∈ ℝ ∧ 𝐵 = +∞) ∨ (𝐴 = -∞ ∧ 𝐵 ∈ ℝ)))))
27	5, 26	orbi12d 794	. 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 8059	. . . 4 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
29	28	breqi 4035	. . 3 ⊢ (𝐴 < 𝐵 ↔ 𝐴({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))𝐵)
30		brun 4080	. . 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 184	. 2 ⊢ (𝐴 < 𝐵 ↔ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ∨ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵))
32		orass 768	. 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 223	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   ∧ wa 104   ↔ wb 105   ∨ wo 709   ∧ w3a 980   = wceq 1364   ∈ wcel 2164   ∪ cun 3151  {csn 3618   class class class wbr 4029  {copab 4089   × cxp 4657  ℝcr 7871   <_ℝ cltrr 7876  +∞cpnf 8051  -∞cmnf 8052  ℝ^*cxr 8053   < clt 8054

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-14 2167  ax-ext 2175  ax-sep 4147  ax-pow 4203  ax-pr 4238

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1472  df-sb 1774  df-eu 2045  df-mo 2046  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-ral 2477  df-rex 2478  df-v 2762  df-un 3157  df-in 3159  df-ss 3166  df-pw 3603  df-sn 3624  df-pr 3625  df-op 3627  df-br 4030  df-opab 4091  df-xp 4665  df-ltxr 8059

This theorem is referenced by:  xrltnr  9845  ltpnf  9846  mnflt  9849  mnfltpnf  9851  pnfnlt  9853  nltmnf  9854