Theorem nnuzdisj

Description: The first N elements of the set of nonnegative integers are distinct from any later members. (Contributed by Glauco Siliprandi, 21-Nov-2020)

	Ref	Expression
Assertion	nnuzdisj	⊢ ( ( 1 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = ∅

Proof

Step	Hyp	Ref	Expression
1		fz1ssfz0	⊢ ( 1 ... 𝑁 ) ⊆ ( 0 ... 𝑁 )
2		ssrin	⊢ ( ( 1 ... 𝑁 ) ⊆ ( 0 ... 𝑁 ) → ( ( 1 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ( ( 0 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) )
3	1 2	ax-mp	⊢ ( ( 1 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ( ( 0 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) )
4		nn0disj	⊢ ( ( 0 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = ∅
5		sseq0	⊢ ( ( ( ( 1 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ⊆ ( ( 0 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) ∧ ( ( 0 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = ∅ ) → ( ( 1 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = ∅ )
6	3 4 5	mp2an	⊢ ( ( 1 ... 𝑁 ) ∩ ( ℤ≥ ‘ ( 𝑁 + 1 ) ) ) = ∅