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Metamath Proof Explorer

Theorem uzssd

Description: Subset relationship for two sets of upper integers. (Contributed by Glauco Siliprandi, 23-Oct-2021)

Ref Expression
Hypothesis uzssd.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
Assertion uzssd ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) )

Proof

Step Hyp Ref Expression
1 uzssd.1 ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
2 uzss ⊢ ( 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) )
3 1 2 syl ⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) )