Theorem sdomne0

Description: A class that strictly dominates any set is not empty. (Suggested by SN, 14-Jan-2025.) (Contributed by RP, 14-Jan-2025)

	Ref	Expression
Assertion	sdomne0	⊢ ( 𝐵 ≺ 𝐴 → 𝐴 ≠ ∅ )

Proof

Step	Hyp	Ref	Expression
1		relsdom	⊢ Rel ≺
2	1	brrelex1i	⊢ ( 𝐵 ≺ 𝐴 → 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	⊢ ( 𝐵 ≺ 𝐴 → ( 𝐵 ≺ 𝐴 → ∅ ≺ 𝐴 ) )
12	1	brrelex2i	⊢ ( ∅ ≺ 𝐴 → 𝐴 ∈ V )
13		0sdomg	⊢ ( 𝐴 ∈ V → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )
14	12 13	syl	⊢ ( ∅ ≺ 𝐴 → ( ∅ ≺ 𝐴 ↔ 𝐴 ≠ ∅ ) )
15	14	ibi	⊢ ( ∅ ≺ 𝐴 → 𝐴 ≠ ∅ )
16	11 15	syl6	⊢ ( 𝐵 ≺ 𝐴 → ( 𝐵 ≺ 𝐴 → 𝐴 ≠ ∅ ) )
17	16	pm2.43i	⊢ ( 𝐵 ≺ 𝐴 → 𝐴 ≠ ∅ )