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Metamath Proof Explorer

Theorem spansneleqi

Description: Membership relation implied by equality of spans. (Contributed by NM, 6-Jun-2004) (New usage is discouraged.)

Ref Expression
Assertion spansneleqi ⊢ A ∈ ℋ → span ⁡ A = span ⁡ B → A ∈ span ⁡ B

Proof

Step Hyp Ref Expression
1 spansnid ⊢ A ∈ ℋ → A ∈ span ⁡ A
2 eleq2 ⊢ span ⁡ A = span ⁡ B → A ∈ span ⁡ A ↔ A ∈ span ⁡ B
3 1 2 syl5ibcom ⊢ A ∈ ℋ → span ⁡ A = span ⁡ B → A ∈ span ⁡ B