Theorem prtlem18

Description: Lemma for prter2 . (Contributed by Rodolfo Medina, 15-Oct-2010) (Revised by Mario Carneiro, 12-Aug-2015)

	Ref	Expression
Hypothesis	prtlem18.1	⊢ ∼ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢 ) }
Assertion	prtlem18	⊢ ( Prt 𝐴 → ( ( 𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣 ) → ( 𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤 ) ) )

Proof

Step	Hyp	Ref	Expression
1		prtlem18.1	⊢ ∼ = { ⟨ 𝑥 , 𝑦 ⟩ ∣ ∃ 𝑢 ∈ 𝐴 ( 𝑥 ∈ 𝑢 ∧ 𝑦 ∈ 𝑢 ) }
2		rspe	⊢ ( ( 𝑣 ∈ 𝐴 ∧ ( 𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣 ) ) → ∃ 𝑣 ∈ 𝐴 ( 𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣 ) )
3	2	expr	⊢ ( ( 𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣 ) → ( 𝑤 ∈ 𝑣 → ∃ 𝑣 ∈ 𝐴 ( 𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣 ) ) )
4	1	prtlem13	⊢ ( 𝑧 ∼ 𝑤 ↔ ∃ 𝑣 ∈ 𝐴 ( 𝑧 ∈ 𝑣 ∧ 𝑤 ∈ 𝑣 ) )
5	3 4	imbitrrdi	⊢ ( ( 𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣 ) → ( 𝑤 ∈ 𝑣 → 𝑧 ∼ 𝑤 ) )
6	5	a1i	⊢ ( Prt 𝐴 → ( ( 𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣 ) → ( 𝑤 ∈ 𝑣 → 𝑧 ∼ 𝑤 ) ) )
7	1	prtlem13	⊢ ( 𝑧 ∼ 𝑤 ↔ ∃ 𝑝 ∈ 𝐴 ( 𝑧 ∈ 𝑝 ∧ 𝑤 ∈ 𝑝 ) )
8		prtlem17	⊢ ( Prt 𝐴 → ( ( 𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣 ) → ( ∃ 𝑝 ∈ 𝐴 ( 𝑧 ∈ 𝑝 ∧ 𝑤 ∈ 𝑝 ) → 𝑤 ∈ 𝑣 ) ) )
9	7 8	syl7bi	⊢ ( Prt 𝐴 → ( ( 𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣 ) → ( 𝑧 ∼ 𝑤 → 𝑤 ∈ 𝑣 ) ) )
10	6 9	impbidd	⊢ ( Prt 𝐴 → ( ( 𝑣 ∈ 𝐴 ∧ 𝑧 ∈ 𝑣 ) → ( 𝑤 ∈ 𝑣 ↔ 𝑧 ∼ 𝑤 ) ) )