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Metamath Proof Explorer

Theorem prssd

Description: Deduction version of prssi : A pair of elements of a class is a subset of the class. (Contributed by Glauco Siliprandi, 17-Aug-2020)

Ref Expression
Hypotheses prssd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
prssd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
Assertion prssd ( 𝜑 → { 𝐴 , 𝐵 } ⊆ 𝐶 )

Proof

Step Hyp Ref Expression
1 prssd.1 ⊢ ( 𝜑 → 𝐴 ∈ 𝐶 )
2 prssd.2 ⊢ ( 𝜑 → 𝐵 ∈ 𝐶 )
3 prssi ⊢ ( ( 𝐴 ∈ 𝐶 ∧ 𝐵 ∈ 𝐶 ) → { 𝐴 , 𝐵 } ⊆ 𝐶 )
4 1 2 3 syl2anc ⊢ ( 𝜑 → { 𝐴 , 𝐵 } ⊆ 𝐶 )