Metamath Proof Explorer


Axiom ax-pre-sup

Description: A nonempty, bounded-above set of reals has a supremum. Axiom 22 of 22 for real and complex numbers, justified by Theorem axpre-sup . Note: Normally new proofs would use axsup . (New usage is discouraged.) (Contributed by NM, 13-Oct-2005)

Ref Expression
Assertion ax-pre-sup ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) ) )

Detailed syntax breakdown

Step Hyp Ref Expression
0 cA 𝐴
1 cr
2 0 1 wss 𝐴 ⊆ ℝ
3 c0
4 0 3 wne 𝐴 ≠ ∅
5 vx 𝑥
6 vy 𝑦
7 6 cv 𝑦
8 cltrr <
9 5 cv 𝑥
10 7 9 8 wbr 𝑦 < 𝑥
11 10 6 0 wral 𝑦𝐴 𝑦 < 𝑥
12 11 5 1 wrex 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥
13 2 4 12 w3a ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 )
14 9 7 8 wbr 𝑥 < 𝑦
15 14 wn ¬ 𝑥 < 𝑦
16 15 6 0 wral 𝑦𝐴 ¬ 𝑥 < 𝑦
17 vz 𝑧
18 17 cv 𝑧
19 7 18 8 wbr 𝑦 < 𝑧
20 19 17 0 wrex 𝑧𝐴 𝑦 < 𝑧
21 10 20 wi ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 )
22 21 6 1 wral 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 )
23 16 22 wa ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) )
24 23 5 1 wrex 𝑥 ∈ ℝ ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) )
25 13 24 wi ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦𝐴 𝑦 < 𝑥 ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧𝐴 𝑦 < 𝑧 ) ) )