Theorem onexgt

Description: For any ordinal, there is always a larger ordinal. (Contributed by RP, 1-Feb-2025)

	Ref	Expression
Assertion	onexgt	⊢ ( 𝐴 ∈ On → ∃ 𝑥 ∈ On 𝐴 ∈ 𝑥 )

Proof

Step	Hyp	Ref	Expression
1		onsuc	⊢ ( 𝐴 ∈ On → suc 𝐴 ∈ On )
2		sucidg	⊢ ( 𝐴 ∈ On → 𝐴 ∈ suc 𝐴 )
3		eleq2	⊢ ( 𝑥 = suc 𝐴 → ( 𝐴 ∈ 𝑥 ↔ 𝐴 ∈ suc 𝐴 ) )
4	3	rspcev	⊢ ( ( suc 𝐴 ∈ On ∧ 𝐴 ∈ suc 𝐴 ) → ∃ 𝑥 ∈ On 𝐴 ∈ 𝑥 )
5	1 2 4	syl2anc	⊢ ( 𝐴 ∈ On → ∃ 𝑥 ∈ On 𝐴 ∈ 𝑥 )