Theorem ltrelxr 7825

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 7805	. 2 ⊢ < = ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)))
2		df-3an 964	. . . . . 6 ⊢ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦) ↔ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦))
3	2	opabbii 3995	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} = {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)}
4		opabssxp 4613	. . . . 5 ⊢ {⟨𝑥, 𝑦⟩ ∣ ((𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ) ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
5	3, 4	eqsstri 3129	. . . 4 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ × ℝ)
6		rexpssxrxp 7810	. . . 4 ⊢ (ℝ × ℝ) ⊆ (ℝ* × ℝ*)
7	5, 6	sstri 3106	. . 3 ⊢ {⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ⊆ (ℝ* × ℝ*)
8		ressxr 7809	. . . . . 6 ⊢ ℝ ⊆ ℝ*
9		snsspr2 3669	. . . . . . 7 ⊢ {-∞} ⊆ {+∞, -∞}
10		ssun2 3240	. . . . . . . 8 ⊢ {+∞, -∞} ⊆ (ℝ ∪ {+∞, -∞})
11		df-xr 7804	. . . . . . . 8 ⊢ ℝ* = (ℝ ∪ {+∞, -∞})
12	10, 11	sseqtrri 3132	. . . . . . 7 ⊢ {+∞, -∞} ⊆ ℝ*
13	9, 12	sstri 3106	. . . . . 6 ⊢ {-∞} ⊆ ℝ*
14	8, 13	unssi 3251	. . . . 5 ⊢ (ℝ ∪ {-∞}) ⊆ ℝ*
15		snsspr1 3668	. . . . . 6 ⊢ {+∞} ⊆ {+∞, -∞}
16	15, 12	sstri 3106	. . . . 5 ⊢ {+∞} ⊆ ℝ*
17		xpss12 4646	. . . . 5 ⊢ (((ℝ ∪ {-∞}) ⊆ ℝ* ∧ {+∞} ⊆ ℝ*) → ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*))
18	14, 16, 17	mp2an 422	. . . 4 ⊢ ((ℝ ∪ {-∞}) × {+∞}) ⊆ (ℝ* × ℝ*)
19		xpss12 4646	. . . . 5 ⊢ (({-∞} ⊆ ℝ* ∧ ℝ ⊆ ℝ*) → ({-∞} × ℝ) ⊆ (ℝ* × ℝ*))
20	13, 8, 19	mp2an 422	. . . 4 ⊢ ({-∞} × ℝ) ⊆ (ℝ* × ℝ*)
21	18, 20	unssi 3251	. . 3 ⊢ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ)) ⊆ (ℝ* × ℝ*)
22	7, 21	unssi 3251	. 2 ⊢ ({⟨𝑥, 𝑦⟩ ∣ (𝑥 ∈ ℝ ∧ 𝑦 ∈ ℝ ∧ 𝑥 <ℝ 𝑦)} ∪ (((ℝ ∪ {-∞}) × {+∞}) ∪ ({-∞} × ℝ))) ⊆ (ℝ* × ℝ*)
23	1, 22	eqsstri 3129	1 ⊢ < ⊆ (ℝ* × ℝ*)

Syntax hints:   ∧ wa 103   ∧ w3a 962   ∈ wcel 1480   ∪ cun 3069   ⊆ wss 3071  {csn 3527  {cpr 3528   class class class wbr 3929  {copab 3988   × cxp 4537  ℝcr 7619   <_ℝ cltrr 7624  +∞cpnf 7797  -∞cmnf 7798  ℝ^*cxr 7799   < clt 7800

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-io 698  ax-5 1423  ax-7 1424  ax-gen 1425  ax-ie1 1469  ax-ie2 1470  ax-8 1482  ax-10 1483  ax-11 1484  ax-i12 1485  ax-bndl 1486  ax-4 1487  ax-17 1506  ax-i9 1510  ax-ial 1514  ax-i5r 1515  ax-ext 2121

This theorem depends on definitions:  df-bi 116  df-3an 964  df-tru 1334  df-nf 1437  df-sb 1736  df-clab 2126  df-cleq 2132  df-clel 2135  df-nfc 2270  df-v 2688  df-un 3075  df-in 3077  df-ss 3084  df-pr 3534  df-opab 3990  df-xp 4545  df-xr 7804  df-ltxr 7805

This theorem is referenced by:  ltrel  7826