Theorem ssneld 3928

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 3925	. 2 ⊢ (𝜑 → (𝐶 ∈ 𝐴 → 𝐶 ∈ 𝐵))
3	2	con3d 152	1 ⊢ (𝜑 → (¬ 𝐶 ∈ 𝐵 → ¬ 𝐶 ∈ 𝐴))

Syntax hints:  ¬ wn 3   → wi 4   ∈ wcel 2104   ⊆ wss 3892

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1911  ax-6 1969  ax-7 2009  ax-8 2106  ax-9 2114  ax-ext 2707

This theorem depends on definitions:  df-bi 206  df-an 398  df-tru 1542  df-ex 1780  df-sb 2066  df-clab 2714  df-cleq 2728  df-clel 2814  df-v 3439  df-in 3899  df-ss 3909

This theorem is referenced by:  ssneldd  3929  kmlem2  9957  hashbclem  14213  prodss  15706  coprmproddvdslem  16416  mrissmrid  17399  mpfrcl  21344  onsuct0  34679  ftc1anc  35906  dvhdimlem  39658  dvh3dim2  39662  dvh3dim3N  39663  mapdh9a  40003  hdmapval0  40047  hdmap11lem2  40056  iundjiunlem  44227  elbigolo1  46147