Theorem nelcon3d 3069

Description: Contrapositive law deduction for negated membership. (Contributed by AV, 28-Jan-2020.)

Hypothesis

Ref	Expression
nelcon3d.1	⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐶 ∈ 𝐷))

Assertion

Ref	Expression
nelcon3d	⊢ (𝜑 → (𝐶 ∉ 𝐷 → 𝐴 ∉ 𝐵))

Proof of Theorem nelcon3d

Step	Hyp	Ref	Expression
1		nelcon3d.1	. . 3 ⊢ (𝜑 → (𝐴 ∈ 𝐵 → 𝐶 ∈ 𝐷))
2	1	con3d 153	. 2 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐷 → ¬ 𝐴 ∈ 𝐵))
3		df-nel 3066	. 2 ⊢ (𝐶 ∉ 𝐷 ↔ ¬ 𝐶 ∈ 𝐷)
4		df-nel 3066	. 2 ⊢ (𝐴 ∉ 𝐵 ↔ ¬ 𝐴 ∈ 𝐵)
5	2, 3, 4	3imtr4g 299	1 ⊢ (𝜑 → (𝐶 ∉ 𝐷 → 𝐴 ∉ 𝐵))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147   ∉ wnel 3065

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8

This proof depends on definitions:  df-bi 210  df-nel 3066

This theorem is used by:  prcssprc  5293  fsetprcnex  8909  lcmfnnval  16825  chninf  18835  isnmgm  18846  mgmplusfreseq  49465