Theorem elunnel1 4149

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

Assertion

Ref	Expression
elunnel1	⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Proof of Theorem elunnel1

Step	Hyp	Ref	Expression
1		elun 4148	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
2	1	biimpi 215	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
3	2	orcanai 1001	1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 396   ∨ wo 845   ∈ wcel 2106   ∪ cun 3946

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2703

This theorem depends on definitions:  df-bi 206  df-an 397  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2710  df-cleq 2724  df-clel 2810  df-v 3476  df-un 3953

This theorem is referenced by:  fsumsplitsn  15692  fprodsplitsn  15935  founiiun0  43968  infxrpnf  44235  cnrefiisplem  44624  dvnprodlem1  44741  fourierdlem70  44971  fourierdlem71  44972  fourierdlem80  44981  sge0splitmpt  45206  sge0iunmptlemfi  45208  nnfoctbdjlem  45250  hoidmvlelem2  45391  hoidmvlelem3  45392  pimrecltpos  45503