Theorem ltrelxr 8087

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 8066	. 2 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2		df-3an 982	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 4100	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4		opabssxp 4737	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
5	3, 4	eqsstri 3215	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
6		rexpssxrxp 8071	. . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7	5, 6	sstri 3192	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ* × ℝ*)
8		ressxr 8070	. . . . . 6 ⊢ ℝ ⊆ ℝ*
9		snsspr2 3771	. . . . . . 7 ⊢ {-∞} ⊆ {+∞, -∞}
10		ssun2 3327	. . . . . . . 8 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11		df-xr 8065	. . . . . . . 8 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
12	10, 11	sseqtrri 3218	. . . . . . 7 ⊢ {+∞, -∞} ⊆ ℝ*
13	9, 12	sstri 3192	. . . . . 6 ⊢ {-∞} ⊆ ℝ*
14	8, 13	unssi 3338	. . . . 5 ⊢ (ℝ ∪ {-∞}) ⊆ ℝ*
15		snsspr1 3770	. . . . . 6 ⊢ {+∞} ⊆ {+∞, -∞}
16	15, 12	sstri 3192	. . . . 5 ⊢ {+∞} ⊆ ℝ*
17		xpss12 4770	. . . . 5 ⊢ (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
18	14, 16, 17	mp2an 426	. . . 4 ⊢ ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19		xpss12 4770	. . . . 5 ⊢ (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
20	13, 8, 19	mp2an 426	. . . 4 ⊢ ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
21	18, 20	unssi 3338	. . 3 ⊢ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
22	7, 21	unssi 3338	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
23	1, 22	eqsstri 3215	1 ⊢ < ⊆ (ℝ* × ℝ*)

Syntax hints:   ∧ wa 104   ∧ w3a 980   ∈ wcel 2167   ∪ cun 3155   ⊆ wss 3157  {csn 3622  {cpr 3623   class class class wbr 4033  {copab 4093   × cxp 4661  ℝcr 7878   <_ℝ cltrr 7883  +∞cpnf 8058  -∞cmnf 8059  ℝ^*cxr 8060   < clt 8061

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 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-3an 982  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-in 3163  df-ss 3170  df-pr 3629  df-opab 4095  df-xp 4669  df-xr 8065  df-ltxr 8066

This theorem is referenced by:  ltrel  8088