Theorem sdomne0d

Description: A class that strictly dominates any set is not empty. (Contributed by RP, 3-Sep-2024)

	Ref	Expression
Hypotheses	sdomne0d.a	⊢ ( 𝜑 → 𝐵 ≺ 𝐴 )
	sdomne0d.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
Assertion	sdomne0d	⊢ ( 𝜑 → 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		sdomne0d.a	⊢ ( 𝜑 → 𝐵 ≺ 𝐴 )
2		sdomne0d.b	⊢ ( 𝜑 → 𝐵 ∈ 𝑉 )
3		breq1	⊢ ( 𝐵 = ∅ → ( 𝐵 ≺ 𝐴 ↔ ∅ ≺ 𝐴 ) )
4	3	biimpd	⊢ ( 𝐵 = ∅ → ( 𝐵 ≺ 𝐴 → ∅ ≺ 𝐴 ) )
5	4	a1i	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 = ∅ → ( 𝐵 ≺ 𝐴 → ∅ ≺ 𝐴 ) ) )
6		0sdomg	⊢ ( 𝐵 ∈ 𝑉 → ( ∅ ≺ 𝐵 ↔ 𝐵 ≠ ∅ ) )
7		sdomtr	⊢ ( ( ∅ ≺ 𝐵 ∧ 𝐵 ≺ 𝐴 ) → ∅ ≺ 𝐴 )
8	7	ex	⊢ ( ∅ ≺ 𝐵 → ( 𝐵 ≺ 𝐴 → ∅ ≺ 𝐴 ) )
9	6 8	biimtrrdi	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ≠ ∅ → ( 𝐵 ≺ 𝐴 → ∅ ≺ 𝐴 ) ) )
10	5 9	pm2.61dne	⊢ ( 𝐵 ∈ 𝑉 → ( 𝐵 ≺ 𝐴 → ∅ ≺ 𝐴 ) )
11	2 10	syl	⊢ ( 𝜑 → ( 𝐵 ≺ 𝐴 → ∅ ≺ 𝐴 ) )
12		relsdom	⊢ Rel ≺
13	12	brrelex2i	⊢ ( ∅ ≺ 𝐴 → 𝐴 ∈ V )
14		0sdomg	⊢ ( 𝐴 ∈ V → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )
15	13 14	syl	⊢ ( ∅ ≺ 𝐴 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )
16	15	ibi	⊢ ( ∅ ≺ 𝐴 → 𝐴 ≠ ∅ )
17	11 16	syl6	⊢ ( 𝜑 → ( 𝐵 ≺ 𝐴 → 𝐴 ≠ ∅ ) )
18	1 17	mpd	⊢ ( 𝜑 → 𝐴 ≠ ∅ )