Theorem ltrelxr 10099

Description: 'Less than' is a relation on extended reals. (Contributed by Mario Carneiro, 28-Apr-2015.)

Assertion

Ref	Expression
ltrelxr	⊢ < ⊆ (ℝ* × ℝ*)

Proof of Theorem ltrelxr

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

Step	Hyp	Ref	Expression
1		df-ltxr 10079	. 2 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2		df-3an 1039	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 4717	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4		opabssxp 5193	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
5	3, 4	eqsstri 3635	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
6		rexpssxrxp 10084	. . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7	5, 6	sstri 3612	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ* × ℝ*)
8		ressxr 10083	. . . . . 6 ⊢ ℝ ⊆ ℝ*
9		snsspr2 4346	. . . . . . 7 ⊢ {-∞} ⊆ {+∞, -∞}
10		ssun2 3777	. . . . . . . 8 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11		df-xr 10078	. . . . . . . 8 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
12	10, 11	sseqtr4i 3638	. . . . . . 7 ⊢ {+∞, -∞} ⊆ ℝ*
13	9, 12	sstri 3612	. . . . . 6 ⊢ {-∞} ⊆ ℝ*
14	8, 13	unssi 3788	. . . . 5 ⊢ (ℝ ∪ {-∞}) ⊆ ℝ*
15		snsspr1 4345	. . . . . 6 ⊢ {+∞} ⊆ {+∞, -∞}
16	15, 12	sstri 3612	. . . . 5 ⊢ {+∞} ⊆ ℝ*
17		xpss12 5225	. . . . 5 ⊢ (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
18	14, 16, 17	mp2an 708	. . . 4 ⊢ ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19		xpss12 5225	. . . . 5 ⊢ (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
20	13, 8, 19	mp2an 708	. . . 4 ⊢ ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
21	18, 20	unssi 3788	. . 3 ⊢ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
22	7, 21	unssi 3788	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
23	1, 22	eqsstri 3635	1 ⊢ < ⊆ (ℝ* × ℝ*)

Syntax hints:   ∧ wa 384   ∧ w3a 1037   ∈ wcel 1990   ∪ cun 3572   ⊆ wss 3574  {csn 4177  {cpr 4179   class class class wbr 4653  {copab 4712   × cxp 5112  ℝcr 9935   <_ℝ cltrr 9940  +∞cpnf 10071  -∞cmnf 10072  ℝ^*cxr 10073   < clt 10074

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1722  ax-4 1737  ax-5 1839  ax-6 1888  ax-7 1935  ax-9 1999  ax-10 2019  ax-11 2034  ax-12 2047  ax-13 2246  ax-ext 2602

This theorem depends on definitions:  df-bi 197  df-or 385  df-an 386  df-3an 1039  df-tru 1486  df-ex 1705  df-nf 1710  df-sb 1881  df-clab 2609  df-cleq 2615  df-clel 2618  df-nfc 2753  df-v 3202  df-un 3579  df-in 3581  df-ss 3588  df-pr 4180  df-opab 4713  df-xp 5120  df-xr 10078  df-ltxr 10079

This theorem is referenced by:  ltrel  10100  dfle2  11980  dflt2  11981  itg2gt0cn  33465