Theorem elunnel1 4094

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 4093	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
2	1	biimpi 216	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
3	2	orcanai 1005	1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 395   ∨ wo 848   ∈ wcel 2114   ∪ cun 3887

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 2708

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 2715  df-cleq 2728  df-clel 2811  df-v 3431  df-un 3894

This theorem is referenced by:  fsumsplitsn  15706  fprodsplitsn  15954  founiiun0  45620  infxrpnf  45874  cnrefiisplem  46257  dvnprodlem1  46374  fourierdlem70  46604  fourierdlem71  46605  fourierdlem80  46614  sge0splitmpt  46839  sge0iunmptlemfi  46841  nnfoctbdjlem  46883  hoidmvlelem2  47024  hoidmvlelem3  47025  pimrecltpos  47136