Theorem elunnel2 41668

Description: A member of a union that is not a member of the second class, is a member of the first class. (Contributed by Glauco Siliprandi, 11-Dec-2019.)

Assertion

Ref	Expression
elunnel2	⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵)

Proof of Theorem elunnel2

Step	Hyp	Ref	Expression
1		elun 4076	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
2	1	biimpi 219	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
3	2	orcomd 868	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐶 ∨ 𝐴 ∈ 𝐵))
4	3	orcanai 1000	1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 844   ∈ wcel 2111   ∪ cun 3879

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1797  ax-4 1811  ax-5 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886

This theorem is referenced by:  limcresiooub  42284  limcresioolb  42285  fourierdlem48  42796  fourierdlem49  42797  fourierdlem101  42849  prsal  42960  isomenndlem  43169  hsphoidmvle2  43224  hsphoidmvle  43225