Theorem numthcor

Description: Any set is strictly dominated by some ordinal. (Contributed by NM, 22-Oct-2003)

	Ref	Expression
Assertion	numthcor	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 ∈ On 𝐴 ≺ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		breq1	⊢ ( 𝑦 = 𝐴 → ( 𝑦 ≺ 𝑥 ↔ 𝐴 ≺ 𝑥 ) )
2	1	rexbidv	⊢ ( 𝑦 = 𝐴 → ( ∃ 𝑥 ∈ On 𝑦 ≺ 𝑥 ↔ ∃ 𝑥 ∈ On 𝐴 ≺ 𝑥 ) )
3		vpwex	⊢ 𝒫 𝑦 ∈ V
4	3	numth2	⊢ ∃ 𝑥 ∈ On 𝑥 ≈ 𝒫 𝑦
5		vex	⊢ 𝑦 ∈ V
6	5	canth2	⊢ 𝑦 ≺ 𝒫 𝑦
7		ensym	⊢ ( 𝑥 ≈ 𝒫 𝑦 → 𝒫 𝑦 ≈ 𝑥 )
8		sdomentr	⊢ ( ( 𝑦 ≺ 𝒫 𝑦 ∧ 𝒫 𝑦 ≈ 𝑥 ) → 𝑦 ≺ 𝑥 )
9	6 7 8	sylancr	⊢ ( 𝑥 ≈ 𝒫 𝑦 → 𝑦 ≺ 𝑥 )
10	9	reximi	⊢ ( ∃ 𝑥 ∈ On 𝑥 ≈ 𝒫 𝑦 → ∃ 𝑥 ∈ On 𝑦 ≺ 𝑥 )
11	4 10	ax-mp	⊢ ∃ 𝑥 ∈ On 𝑦 ≺ 𝑥
12	2 11	vtoclg	⊢ ( 𝐴 ∈ 𝑉 → ∃ 𝑥 ∈ On 𝐴 ≺ 𝑥 )