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Metamath Proof Explorer

Theorem dfpss3

Description: Alternate definition of proper subclass. (Contributed by NM, 7-Feb-1996) (Proof shortened by Andrew Salmon, 26-Jun-2011)

Ref Expression
Assertion dfpss3 ⊢ A ⊂ B ↔ A ⊆ B ∧ ¬ B ⊆ A

Proof

Step Hyp Ref Expression
1 dfpss2 ⊢ A ⊂ B ↔ A ⊆ B ∧ ¬ A = B
2 eqss ⊢ A = B ↔ A ⊆ B ∧ B ⊆ A
3 2 baib ⊢ A ⊆ B → A = B ↔ B ⊆ A
4 3 notbid ⊢ A ⊆ B → ¬ A = B ↔ ¬ B ⊆ A
5 4 pm5.32i ⊢ A ⊆ B ∧ ¬ A = B ↔ A ⊆ B ∧ ¬ B ⊆ A
6 1 5 bitri ⊢ A ⊂ B ↔ A ⊆ B ∧ ¬ B ⊆ A