This is an inofficial mirror of http://metamath.tirix.org for personal testing of a visualizer extension only.

Metamath Proof Explorer

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 )