Metamath Proof Explorer

Theorem ordpss

Description: ordelpss with an antecedent removed. (Contributed by Andrew Salmon, 25-Jul-2011)

Ref Expression
Assertion ordpss ⊢ Ord ⁡ B → A ∈ B → A ⊂ B

Proof

Step Hyp Ref Expression
1 ordelord ⊢ Ord ⁡ B ∧ A ∈ B → Ord ⁡ A
2 1 ex ⊢ Ord ⁡ B → A ∈ B → Ord ⁡ A
3 2 ancrd ⊢ Ord ⁡ B → A ∈ B → Ord ⁡ A ∧ A ∈ B
4 ordelpss ⊢ Ord ⁡ A ∧ Ord ⁡ B → A ∈ B ↔ A ⊂ B
5 4 ancoms ⊢ Ord ⁡ B ∧ Ord ⁡ A → A ∈ B ↔ A ⊂ B
6 5 biimpd ⊢ Ord ⁡ B ∧ Ord ⁡ A → A ∈ B → A ⊂ B
7 6 expimpd ⊢ Ord ⁡ B → Ord ⁡ A ∧ A ∈ B → A ⊂ B
8 3 7 syld ⊢ Ord ⁡ B → A ∈ B → A ⊂ B