Theorem elunnel1 4105

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 4104	. . 3 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) ↔ (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
2	1	biimpi 218	. 2 ⊢ (𝐴 ∈ (𝐵 ∪ 𝐶) → (𝐴 ∈ 𝐵 ∨ 𝐴 ∈ 𝐶))
3	2	orcanai 1015	1 ⊢ ((𝐴 ∈ (𝐵 ∪ 𝐶) ∧ ¬ 𝐴 ∈ 𝐵) → 𝐴 ∈ 𝐶)

Syntax hints:  ¬ wn 3   → wi 4   ∧ wa 399   ∨ wo 858   ∈ wcel 2141   ∪ cun 3900

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1814  ax-4 1828  ax-5 1929  ax-6 1986  ax-7 2027  ax-8 2143  ax-9 2151  ax-ext 2733

This theorem depends on definitions:  df-bi 209  df-an 400  df-or 859  df-tru 1562  df-ex 1799  df-sb 2090  df-clab 2740  df-cleq 2753  df-clel 2836  df-v 3455  df-un 3907

This theorem is referenced by:  fsumsplitsn  15762  fprodsplitsn  16010  founiiun0  45729  infxrpnf  45981  cnrefiisplem  46364  dvnprodlem1  46481  fourierdlem70  46711  fourierdlem71  46712  fourierdlem80  46721  sge0splitmpt  46946  sge0iunmptlemfi  46948  nnfoctbdjlem  46990  hoidmvlelem2  47131  hoidmvlelem3  47132  pimrecltpos  47243