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Metamath Proof Explorer

Theorem supnub

Description: An upper bound is not less than the supremum. (Contributed by NM, 13-Oct-2004)

Ref Expression
Hypotheses supmo.1 ⊢ ( 𝜑 → 𝑅 Or 𝐴 )
supcl.2 ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) )
Assertion supnub ( 𝜑 → ( ( 𝐶 ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐵 ¬ 𝐶 𝑅 𝑧 ) → ¬ 𝐶 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) ) )

Proof

Step Hyp Ref Expression
1 supmo.1 ⊢ ( 𝜑 → 𝑅 Or 𝐴 )
2 supcl.2 ⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑥 𝑅 𝑦 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑦 𝑅 𝑥 → ∃ 𝑧 ∈ 𝐵 𝑦 𝑅 𝑧 ) ) )
3 1 2 suplub ⊢ ( 𝜑 → ( ( 𝐶 ∈ 𝐴 ∧ 𝐶 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) ) → ∃ 𝑧 ∈ 𝐵 𝐶 𝑅 𝑧 ) )
4 3 expdimp ⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐶 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) → ∃ 𝑧 ∈ 𝐵 𝐶 𝑅 𝑧 ) )
5 dfrex2 ⊢ ( ∃ 𝑧 ∈ 𝐵 𝐶 𝑅 𝑧 ↔ ¬ ∀ 𝑧 ∈ 𝐵 ¬ 𝐶 𝑅 𝑧 )
6 4 5 imbitrdi ⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ( 𝐶 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) → ¬ ∀ 𝑧 ∈ 𝐵 ¬ 𝐶 𝑅 𝑧 ) )
7 6 con2d ⊢ ( ( 𝜑 ∧ 𝐶 ∈ 𝐴 ) → ( ∀ 𝑧 ∈ 𝐵 ¬ 𝐶 𝑅 𝑧 → ¬ 𝐶 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) ) )
8 7 expimpd ⊢ ( 𝜑 → ( ( 𝐶 ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐵 ¬ 𝐶 𝑅 𝑧 ) → ¬ 𝐶 𝑅 sup ( 𝐵 , 𝐴 , 𝑅 ) ) )