Theorem onpsssuc

Description: An ordinal number is a proper subset of its successor. (Contributed by Stefan O'Rear, 18-Nov-2014)

	Ref	Expression
Assertion	onpsssuc	⊢ ( 𝐴 ∈ On → 𝐴 ⊊ suc 𝐴 )

Proof

Step	Hyp	Ref	Expression
1		sucidg	⊢ ( 𝐴 ∈ On → 𝐴 ∈ suc 𝐴 )
2		eloni	⊢ ( 𝐴 ∈ On → Ord 𝐴 )
3		ordsuc	⊢ ( Ord 𝐴 ↔ Ord suc 𝐴 )
4	2 3	sylib	⊢ ( 𝐴 ∈ On → Ord suc 𝐴 )
5		ordelpss	⊢ ( ( Ord 𝐴 ∧ Ord suc 𝐴 ) → ( 𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴 ) )
6	2 4 5	syl2anc	⊢ ( 𝐴 ∈ On → ( 𝐴 ∈ suc 𝐴 ↔ 𝐴 ⊊ suc 𝐴 ) )
7	1 6	mpbid	⊢ ( 𝐴 ∈ On → 𝐴 ⊊ suc 𝐴 )