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Metamath Proof Explorer

Theorem oppccat

Description: An opposite category is a category. (Contributed by Mario Carneiro, 2-Jan-2017)

Ref Expression
Hypothesis oppcbas.1 ⊢ O = oppCat ⁡ C
Assertion oppccat ⊢ C ∈ Cat → O ∈ Cat

Proof

Step Hyp Ref Expression
1 oppcbas.1 ⊢ O = oppCat ⁡ C
2 1 oppccatid ⊢ C ∈ Cat → O ∈ Cat ∧ Id ⁡ O = Id ⁡ C
3 2 simpld ⊢ C ∈ Cat → O ∈ Cat