Theorem ssneldd 3145

Description: If an element is not in a class, it is also not in a subclass of that class. Deduction form. (Contributed by David Moews, 1-May-2017.)

Hypotheses

Ref	Expression
ssneld.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)
ssneldd.2	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)

Assertion

Ref	Expression
ssneldd	⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)

Proof of Theorem ssneldd

Step	Hyp	Ref	Expression
1		ssneldd.2	. 2 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐵)
2		ssneld.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
3	2	ssneld 3144	. 2 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))
4	1, 3	mpd 13	1 ⊢ (𝜑 → ¬ 𝐶 ∈ 𝐴)

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2136   ⊆ wss 3116

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-in1 604  ax-in2 605  ax-5 1435  ax-7 1436  ax-gen 1437  ax-ie1 1481  ax-ie2 1482  ax-8 1492  ax-11 1494  ax-4 1498  ax-17 1514  ax-i9 1518  ax-ial 1522  ax-i5r 1523  ax-ext 2147

This theorem depends on definitions:  df-bi 116  df-nf 1449  df-sb 1751  df-clab 2152  df-cleq 2158  df-clel 2161  df-in 3122  df-ss 3129

This theorem is referenced by:  0nelrel  4650  addnqprlemfl  7500  addnqprlemfu  7501  mulnqprlemfl  7516  mulnqprlemfu  7517  cauappcvgprlemladdru  7597  fprodntrivap  11525  fprodssdc  11531