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Metamath Proof Explorer

Theorem uzssd2

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

Ref Expression
Hypotheses uzssd2.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
uzssd2.2 ⊢ ( 𝜑 → 𝑁 ∈ 𝑍 )
Assertion uzssd2 ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ 𝑍 )

Proof

Step Hyp Ref Expression
1 uzssd2.1 ⊢ 𝑍 = ( ℤ≥ ‘ 𝑀 )
2 uzssd2.2 ⊢ ( 𝜑 → 𝑁 ∈ 𝑍 )
3 2 1 eleqtrdi ⊢ ( 𝜑 → 𝑁 ∈ ( ℤ≥ ‘ 𝑀 ) )
4 3 uzssd ⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ ( ℤ≥ ‘ 𝑀 ) )
5 4 1 sseqtrrdi ⊢ ( 𝜑 → ( ℤ≥ ‘ 𝑁 ) ⊆ 𝑍 )