Theorem ltxrlt 11183

Description: The standard less-than <_ℝ and the extended real less-than < are identical when restricted to the non-extended reals ℝ. (Contributed by NM, 13-Oct-2005.) (Revised by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltxrlt	⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵))

Proof of Theorem ltxrlt

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

Step	Hyp	Ref	Expression
1		brun 5140	. . . . 5 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ↔ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵))
2		brxp 5663	. . . . . . 7 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ↔ (𝐴 ∈ (ℝ ∪ {-∞}) ∧ 𝐵 ∈ {+∞}))
3		elsni 4590	. . . . . . . 8 ⊢ (𝐵 ∈ {+∞} → 𝐵 = +∞)
4		pnfnre 11153	. . . . . . . . . 10 ⊢ +∞ ∉ ℝ
5	4	neli 3034	. . . . . . . . 9 ⊢ ¬ +∞ ∈ ℝ
6		simpr 484	. . . . . . . . . 10 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐵 ∈ ℝ)
7		eleq1 2819	. . . . . . . . . 10 ⊢ (𝐵 = +∞ → (𝐵 ∈ ℝ ↔ +∞ ∈ ℝ))
8	6, 7	imbitrid 244	. . . . . . . . 9 ⊢ (𝐵 = +∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → +∞ ∈ ℝ))
9	5, 8	mtoi 199	. . . . . . . 8 ⊢ (𝐵 = +∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
10	3, 9	syl 17	. . . . . . 7 ⊢ (𝐵 ∈ {+∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
11	2, 10	simplbiim 504	. . . . . 6 ⊢ (𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
12		brxp 5663	. . . . . . 7 ⊢ (𝐴({-∞} × ℝ)𝐵 ↔ (𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ))
13		elsni 4590	. . . . . . . . 9 ⊢ (𝐴 ∈ {-∞} → 𝐴 = -∞)
14		mnfnre 11155	. . . . . . . . . . 11 ⊢ -∞ ∉ ℝ
15	14	neli 3034	. . . . . . . . . 10 ⊢ ¬ -∞ ∈ ℝ
16		simpl 482	. . . . . . . . . . 11 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → 𝐴 ∈ ℝ)
17		eleq1 2819	. . . . . . . . . . 11 ⊢ (𝐴 = -∞ → (𝐴 ∈ ℝ ↔ -∞ ∈ ℝ))
18	16, 17	imbitrid 244	. . . . . . . . . 10 ⊢ (𝐴 = -∞ → ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → -∞ ∈ ℝ))
19	15, 18	mtoi 199	. . . . . . . . 9 ⊢ (𝐴 = -∞ → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
20	13, 19	syl 17	. . . . . . . 8 ⊢ (𝐴 ∈ {-∞} → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
21	20	adantr 480	. . . . . . 7 ⊢ ((𝐴 ∈ {-∞} ∧ 𝐵 ∈ ℝ) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
22	12, 21	sylbi 217	. . . . . 6 ⊢ (𝐴({-∞} × ℝ)𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
23	11, 22	jaoi 857	. . . . 5 ⊢ ((𝐴((ℝ ∪ {-∞}) × {+∞})𝐵 ∨ 𝐴({-∞} × ℝ)𝐵) → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
24	1, 23	sylbi 217	. . . 4 ⊢ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → ¬ (𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ))
25	24	con2i 139	. . 3 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → ¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵)
26		df-ltxr 11151	. . . . . . 7 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
27	26	equncomi 4107	. . . . . 6 ⊢ < = ((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)})
28	27	breqi 5095	. . . . 5 ⊢ (𝐴 < 𝐵 ↔ 𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)})𝐵)
29		brun 5140	. . . . 5 ⊢ (𝐴((((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ∪ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)})𝐵 ↔ (𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ∨ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
30		df-or 848	. . . . 5 ⊢ ((𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 ∨ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵) ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
31	28, 29, 30	3bitri 297	. . . 4 ⊢ (𝐴 < 𝐵 ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
32		biimt 360	. . . 4 ⊢ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵)))
33	31, 32	bitr4id 290	. . 3 ⊢ (¬ 𝐴(((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))𝐵 → (𝐴 < 𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
34	25, 33	syl 17	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
35		breq12 5094	. . . 4 ⊢ ((𝑥 = 𝐴 ∧ 𝑦 = 𝐵) → (𝑥 <ℝ 𝑦 ↔ 𝐴 <ℝ 𝐵))
36		df-3an 1088	. . . . 5 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
37	36	opabbii 5156	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
38	35, 37	brab2a 5707	. . 3 ⊢ (𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵 ↔ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) ∧ 𝐴 <ℝ 𝐵))
39	38	baibr 536	. 2 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 <ℝ 𝐵 ↔ 𝐴{⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)}𝐵))
40	34, 39	bitr4d 282	1 ⊢ ((𝐴 ∈ ℝ ∧ 𝐵 ∈ ℝ) → (𝐴 < 𝐵 ↔ 𝐴 <ℝ 𝐵))

Syntax hints:  ¬ wn 3   → wi 4   ↔ wb 206   ∧ wa 395   ∨ wo 847   ∧ w3a 1086   = wceq 1541   ∈ wcel 2111   ∪ cun 3895  {csn 4573   class class class wbr 5089  {copab 5151   × cxp 5612  ℝcr 11005   <_ℝ cltrr 11010  +∞cpnf 11143  -∞cmnf 11144   < clt 11146

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 1968  ax-7 2009  ax-8 2113  ax-9 2121  ax-10 2144  ax-11 2160  ax-12 2180  ax-ext 2703  ax-sep 5232  ax-nul 5242  ax-pow 5301  ax-pr 5368  ax-un 7668  ax-resscn 11063

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-3an 1088  df-tru 1544  df-fal 1554  df-ex 1781  df-nf 1785  df-sb 2068  df-mo 2535  df-eu 2564  df-clab 2710  df-cleq 2723  df-clel 2806  df-nfc 2881  df-ne 2929  df-nel 3033  df-ral 3048  df-rex 3057  df-rab 3396  df-v 3438  df-sbc 3737  df-csb 3846  df-dif 3900  df-un 3902  df-in 3904  df-ss 3914  df-nul 4281  df-if 4473  df-pw 4549  df-sn 4574  df-pr 4576  df-op 4580  df-uni 4857  df-br 5090  df-opab 5152  df-mpt 5171  df-id 5509  df-xp 5620  df-rel 5621  df-cnv 5622  df-co 5623  df-dm 5624  df-rn 5625  df-res 5626  df-ima 5627  df-iota 6437  df-fun 6483  df-fn 6484  df-f 6485  df-f1 6486  df-fo 6487  df-f1o 6488  df-fv 6489  df-er 8622  df-en 8870  df-dom 8871  df-sdom 8872  df-pnf 11148  df-mnf 11149  df-ltxr 11151

This theorem is referenced by:  axlttri  11184  axlttrn  11185  axltadd  11186  axmulgt0  11187  axsup  11188