Theorem elunnel2 4109

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 4107	. . . 4 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
2	1	biimpi 216	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
3	2	orcomd 872	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐶 ∨ 𝐴 ∈ 𝐵))
4	3	orcanai 1005	1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐶) → 𝐴 ∈ 𝐵)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   ∈ wcel 2114   ∪ cun 3901

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2709

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2716  df-cleq 2729  df-clel 2812  df-v 3444  df-un 3908

This theorem is referenced by:  fvn0fvelrn  6871  limcresiooub  46000  limcresioolb  46001  fourierdlem48  46512  fourierdlem49  46513  fourierdlem101  46565  prsal  46676  isomenndlem  46888  hsphoidmvle2  46943  hsphoidmvle  46944