Theorem ssneld 3915

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

Hypothesis

Ref	Expression
ssneld.1	⊢ (𝜑 → 𝐴 ⊆ 𝐵)

Assertion

Ref	Expression
ssneld	⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))

Proof of Theorem ssneld

Step	Hyp	Ref	Expression
1		ssneld.1	. . 3 ⊢ (𝜑 → 𝐴 ⊆ 𝐵)
2	1	sseld 3912	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
3	2	con3d 153	1 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))

This proof depends on syntax axioms:  ¬ wn 3   → wi 4   ∈ wcel 2147   ⊆ wss 3900

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1813  df-clel 2839  df-ss 3901

This theorem is used by:  ssneldd  3916  kmlem2  10269  hashbclem  14638  prodss  16158  coprmproddvdslem  16881  mrissmrid  17865  mpfrcl  22449  onsuct0  37444  ftc1anc  38887  dvhdimlem  42767  dvh3dim2  42771  dvh3dim3N  42772  mapdh9a  43112  hdmapval0  43156  hdmap11lem2  43165  iundjiunlem  47742  elbigolo1  50000