Metamath Proof Explorer


Definition df-np

Description: Define the set of positive reals. A "Dedekind cut" is a partition of the positive rational numbers into two classes such that all the numbers of one class are less than all the numbers of the other. A positive real is defined as the lower class of a Dedekind cut. Definition 9-3.1 of Gleason p. 121. (Note: This is a "temporary" definition used in the construction of complex numbers df-c , and is intended to be used only by the construction.) (Contributed by NM, 31-Oct-1995) (New usage is discouraged.)

Ref Expression
Assertion df-np P = { 𝑥 ∣ ( ( ∅ ⊊ 𝑥𝑥Q ) ∧ ∀ 𝑦𝑥 ( ∀ 𝑧 ( 𝑧 <Q 𝑦𝑧𝑥 ) ∧ ∃ 𝑧𝑥 𝑦 <Q 𝑧 ) ) }

Detailed syntax breakdown

Step Hyp Ref Expression
0 cnp P
1 vx 𝑥
2 c0
3 1 cv 𝑥
4 2 3 wpss ∅ ⊊ 𝑥
5 cnq Q
6 3 5 wpss 𝑥Q
7 4 6 wa ( ∅ ⊊ 𝑥𝑥Q )
8 vy 𝑦
9 vz 𝑧
10 9 cv 𝑧
11 cltq <Q
12 8 cv 𝑦
13 10 12 11 wbr 𝑧 <Q 𝑦
14 10 3 wcel 𝑧𝑥
15 13 14 wi ( 𝑧 <Q 𝑦𝑧𝑥 )
16 15 9 wal 𝑧 ( 𝑧 <Q 𝑦𝑧𝑥 )
17 12 10 11 wbr 𝑦 <Q 𝑧
18 17 9 3 wrex 𝑧𝑥 𝑦 <Q 𝑧
19 16 18 wa ( ∀ 𝑧 ( 𝑧 <Q 𝑦𝑧𝑥 ) ∧ ∃ 𝑧𝑥 𝑦 <Q 𝑧 )
20 19 8 3 wral 𝑦𝑥 ( ∀ 𝑧 ( 𝑧 <Q 𝑦𝑧𝑥 ) ∧ ∃ 𝑧𝑥 𝑦 <Q 𝑧 )
21 7 20 wa ( ( ∅ ⊊ 𝑥𝑥Q ) ∧ ∀ 𝑦𝑥 ( ∀ 𝑧 ( 𝑧 <Q 𝑦𝑧𝑥 ) ∧ ∃ 𝑧𝑥 𝑦 <Q 𝑧 ) )
22 21 1 cab { 𝑥 ∣ ( ( ∅ ⊊ 𝑥𝑥Q ) ∧ ∀ 𝑦𝑥 ( ∀ 𝑧 ( 𝑧 <Q 𝑦𝑧𝑥 ) ∧ ∃ 𝑧𝑥 𝑦 <Q 𝑧 ) ) }
23 0 22 wceq P = { 𝑥 ∣ ( ( ∅ ⊊ 𝑥𝑥Q ) ∧ ∀ 𝑦𝑥 ( ∀ 𝑧 ( 𝑧 <Q 𝑦𝑧𝑥 ) ∧ ∃ 𝑧𝑥 𝑦 <Q 𝑧 ) ) }