Theorem ssneld 3911

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 3908	. 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 2144   ⊆ wss 3896

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1827  ax-4 1841  ax-5 1942  ax-6 1999  ax-7 2040  ax-8 2146

This proof depends on definitions:  df-bi 210  df-an 402  df-ex 1812  df-clel 2835  df-ss 3897

This theorem is used by:  ssneldd  3912  kmlem2  10235  hashbclem  14602  prodss  16121  coprmproddvdslem  16844  mrissmrid  17822  mpfrcl  22401  onsuct0  37323  ftc1anc  38726  dvhdimlem  42608  dvh3dim2  42612  dvh3dim3N  42613  mapdh9a  42953  hdmapval0  42997  hdmap11lem2  43006  iundjiunlem  47576  elbigolo1  49776