Theorem rencldnfi

Description: A set of real numbers which comes arbitrarily close to some target yet excludes it is infinite. The work is done in rencldnfilem using infima; this theorem removes the requirement that A be nonempty. (Contributed by Stefan O'Rear, 19-Oct-2014)

	Ref	Expression
Assertion	rencldnfi	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → ¬ 𝐴 ∈ Fin )

Proof

Step	Hyp	Ref	Expression
1		simpl1	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → 𝐴 ⊆ ℝ )
2		simpl2	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → 𝐵 ∈ ℝ )
3		rexn0	⊢ ( ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 → 𝐴 ≠ ∅ )
4	3	ralimi	⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 → ∀ 𝑥 ∈ ℝ+ 𝐴 ≠ ∅ )
5		1rp	⊢ 1 ∈ ℝ+
6		ne0i	⊢ ( 1 ∈ ℝ+ → ℝ+ ≠ ∅ )
7		r19.3rzv	⊢ ( ℝ+ ≠ ∅ → ( 𝐴 ≠ ∅ ↔ ∀ 𝑥 ∈ ℝ+ 𝐴 ≠ ∅ ) )
8	5 6 7	mp2b	⊢ ( 𝐴 ≠ ∅ ↔ ∀ 𝑥 ∈ ℝ+ 𝐴 ≠ ∅ )
9	4 8	sylibr	⊢ ( ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 → 𝐴 ≠ ∅ )
10	9	adantl	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → 𝐴 ≠ ∅ )
11		simpl3	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → ¬ 𝐵 ∈ 𝐴 )
12	10 11	jca	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → ( 𝐴 ≠ ∅ ∧ ¬ 𝐵 ∈ 𝐴 ) )
13		simpr	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 )
14		rencldnfilem	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ( 𝐴 ≠ ∅ ∧ ¬ 𝐵 ∈ 𝐴 ) ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → ¬ 𝐴 ∈ Fin )
15	1 2 12 13 14	syl31anc	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝐵 ∈ ℝ ∧ ¬ 𝐵 ∈ 𝐴 ) ∧ ∀ 𝑥 ∈ ℝ+ ∃ 𝑦 ∈ 𝐴 ( abs ‘ ( 𝑦 − 𝐵 ) ) < 𝑥 ) → ¬ 𝐴 ∈ Fin )