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Metamath Proof Explorer

Theorem z2ge

Description: There exists an integer greater than or equal to any two others. (Contributed by NM, 28-Aug-2005)

Ref Expression
Assertion z2ge ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ∃ 𝑘 ∈ ℤ ( 𝑀 ≤ 𝑘 ∧ 𝑁 ≤ 𝑘 ) )

Proof

Step Hyp Ref Expression
1 ifcl ⊢ ( ( 𝑁 ∈ ℤ ∧ 𝑀 ∈ ℤ ) → if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℤ )
2 1 ancoms ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℤ )
3 zre ⊢ ( 𝑀 ∈ ℤ → 𝑀 ∈ ℝ )
4 zre ⊢ ( 𝑁 ∈ ℤ → 𝑁 ∈ ℝ )
5 max1 ⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) )
6 max2 ⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) )
7 5 6 jca ⊢ ( ( 𝑀 ∈ ℝ ∧ 𝑁 ∈ ℝ ) → ( 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∧ 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) )
8 3 4 7 syl2an ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ( 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∧ 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) )
9 breq2 ⊢ ( 𝑘 = if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) → ( 𝑀 ≤ 𝑘 ↔ 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) )
10 breq2 ⊢ ( 𝑘 = if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) → ( 𝑁 ≤ 𝑘 ↔ 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) )
11 9 10 anbi12d ⊢ ( 𝑘 = if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) → ( ( 𝑀 ≤ 𝑘 ∧ 𝑁 ≤ 𝑘 ) ↔ ( 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∧ 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) )
12 11 rspcev ⊢ ( ( if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∈ ℤ ∧ ( 𝑀 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ∧ 𝑁 ≤ if ( 𝑀 ≤ 𝑁 , 𝑁 , 𝑀 ) ) ) → ∃ 𝑘 ∈ ℤ ( 𝑀 ≤ 𝑘 ∧ 𝑁 ≤ 𝑘 ) )
13 2 8 12 syl2anc ⊢ ( ( 𝑀 ∈ ℤ ∧ 𝑁 ∈ ℤ ) → ∃ 𝑘 ∈ ℤ ( 𝑀 ≤ 𝑘 ∧ 𝑁 ≤ 𝑘 ) )