Theorem ssneld 3971

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 3968	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
3	2	con3d 155	1 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2114   ⊆ wss 3938

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 1970  ax-7 2015  ax-8 2116  ax-9 2124  ax-10 2145  ax-11 2161  ax-12 2177  ax-ext 2795

This theorem depends on definitions:  df-bi 209  df-an 399  df-or 844  df-tru 1540  df-ex 1781  df-nf 1785  df-sb 2070  df-clab 2802  df-cleq 2816  df-clel 2895  df-in 3945  df-ss 3954

This theorem is referenced by:  ssneldd  3972  kmlem2  9579  hashbclem  13813  prodss  15303  coprmproddvdslem  16008  mrissmrid  16914  mpfrcl  20300  onsuct0  33791  ftc1anc  34977  dvhdimlem  38582  dvh3dim2  38586  dvh3dim3N  38587  mapdh9a  38927  hdmapval0  38971  hdmap11lem2  38980  iundjiunlem  42748  elbigolo1  44624