Theorem sn-sup3d

Description: sup3 without ax-mulcom , proven trivially from sn-sup2 . (Contributed by SN, 29-Jun-2025)

	Ref	Expression
Hypotheses	sn-sup3d.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
	sn-sup3d.2	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
	sn-sup3d.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
Assertion	sn-sup3d	⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) )

Proof

Step	Hyp	Ref	Expression
1		sn-sup3d.1	⊢ ( 𝜑 → 𝐴 ⊆ ℝ )
2		sn-sup3d.2	⊢ ( 𝜑 → 𝐴 ≠ ∅ )
3		sn-sup3d.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 )
4		ssel	⊢ ( 𝐴 ⊆ ℝ → ( 𝑦 ∈ 𝐴 → 𝑦 ∈ ℝ ) )
5		leloe	⊢ ( ( 𝑦 ∈ ℝ ∧ 𝑥 ∈ ℝ ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) )
6	5	expcom	⊢ ( 𝑥 ∈ ℝ → ( 𝑦 ∈ ℝ → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) ) )
7	4 6	syl9	⊢ ( 𝐴 ⊆ ℝ → ( 𝑥 ∈ ℝ → ( 𝑦 ∈ 𝐴 → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) ) ) )
8	7	imp31	⊢ ( ( ( 𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ ) ∧ 𝑦 ∈ 𝐴 ) → ( 𝑦 ≤ 𝑥 ↔ ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) )
9	8	ralbidva	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝑥 ∈ ℝ ) → ( ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∀ 𝑦 ∈ 𝐴 ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) )
10	9	rexbidva	⊢ ( 𝐴 ⊆ ℝ → ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) )
11	1 10	syl	⊢ ( 𝜑 → ( ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 𝑦 ≤ 𝑥 ↔ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) )
12	3 11	mpbid	⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) )
13		sn-sup2	⊢ ( ( 𝐴 ⊆ ℝ ∧ 𝐴 ≠ ∅ ∧ ∃ 𝑥 ∈ ℝ ∀ 𝑦 ∈ 𝐴 ( 𝑦 < 𝑥 ∨ 𝑦 = 𝑥 ) ) → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) )
14	1 2 12 13	syl3anc	⊢ ( 𝜑 → ∃ 𝑥 ∈ ℝ ( ∀ 𝑦 ∈ 𝐴 ¬ 𝑥 < 𝑦 ∧ ∀ 𝑦 ∈ ℝ ( 𝑦 < 𝑥 → ∃ 𝑧 ∈ 𝐴 𝑦 < 𝑧 ) ) )