Theorem ssneld 3931

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 3928	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
3	2	con3d 152	1 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2111   ⊆ wss 3897

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1796  ax-4 1810  ax-5 1911  ax-6 1968  ax-7 2009  ax-8 2113

This theorem depends on definitions:  df-bi 207  df-an 396  df-ex 1781  df-clel 2806  df-ss 3914

This theorem is referenced by:  ssneldd  3932  kmlem2  10038  hashbclem  14354  prodss  15849  coprmproddvdslem  16568  mrissmrid  17542  mpfrcl  22015  onsuct0  36475  ftc1anc  37741  dvhdimlem  41483  dvh3dim2  41487  dvh3dim3N  41488  mapdh9a  41828  hdmapval0  41872  hdmap11lem2  41881  iundjiunlem  46497  elbigolo1  48589