Theorem infssd

Description: Inequality deduction for infimum of a subset. (Contributed by AV, 4-Oct-2020)

	Ref	Expression
Hypotheses	infssd.0	⊢ ( 𝜑 → 𝑅 Or 𝐴 )
	infssd.1	⊢ ( 𝜑 → 𝐶 ⊆ 𝐵 )
	infssd.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐶 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐶 𝑧 𝑅 𝑦 ) ) )
	infssd.4	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ) )
Assertion	infssd	⊢ ( 𝜑 → ¬ inf ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 inf ( 𝐵 , 𝐴 , 𝑅 ) )

Proof

Step	Hyp	Ref	Expression
1		infssd.0	⊢ ( 𝜑 → 𝑅 Or 𝐴 )
2		infssd.1	⊢ ( 𝜑 → 𝐶 ⊆ 𝐵 )
3		infssd.3	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐶 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐶 𝑧 𝑅 𝑦 ) ) )
4		infssd.4	⊢ ( 𝜑 → ∃ 𝑥 ∈ 𝐴 ( ∀ 𝑦 ∈ 𝐵 ¬ 𝑦 𝑅 𝑥 ∧ ∀ 𝑦 ∈ 𝐴 ( 𝑥 𝑅 𝑦 → ∃ 𝑧 ∈ 𝐵 𝑧 𝑅 𝑦 ) ) )
5	1 4	infcl	⊢ ( 𝜑 → inf ( 𝐵 , 𝐴 , 𝑅 ) ∈ 𝐴 )
6	2	sseld	⊢ ( 𝜑 → ( 𝑧 ∈ 𝐶 → 𝑧 ∈ 𝐵 ) )
7	1 4	inflb	⊢ ( 𝜑 → ( 𝑧 ∈ 𝐵 → ¬ 𝑧 𝑅 inf ( 𝐵 , 𝐴 , 𝑅 ) ) )
8	6 7	syld	⊢ ( 𝜑 → ( 𝑧 ∈ 𝐶 → ¬ 𝑧 𝑅 inf ( 𝐵 , 𝐴 , 𝑅 ) ) )
9	8	ralrimiv	⊢ ( 𝜑 → ∀ 𝑧 ∈ 𝐶 ¬ 𝑧 𝑅 inf ( 𝐵 , 𝐴 , 𝑅 ) )
10	1 3	infnlb	⊢ ( 𝜑 → ( ( inf ( 𝐵 , 𝐴 , 𝑅 ) ∈ 𝐴 ∧ ∀ 𝑧 ∈ 𝐶 ¬ 𝑧 𝑅 inf ( 𝐵 , 𝐴 , 𝑅 ) ) → ¬ inf ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 inf ( 𝐵 , 𝐴 , 𝑅 ) ) )
11	5 9 10	mp2and	⊢ ( 𝜑 → ¬ inf ( 𝐶 , 𝐴 , 𝑅 ) 𝑅 inf ( 𝐵 , 𝐴 , 𝑅 ) )