Theorem bnj1424 35235

Description: First-order logic and set theory. (Contributed by Jonathan Ben-Naim, 3-Jun-2011.) (New usage is discouraged.)

Hypothesis

Ref	Expression
bnj1424.1	⊢ 𝐴 = (𝐵 ∪ 𝐶)

Assertion

Ref	Expression
bnj1424	⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))

Proof of Theorem bnj1424

Step	Hyp	Ref	Expression
1		bnj1424.1	. . 3 ⊢ 𝐴 = (𝐵 ∪ 𝐶)
2	1	bnj1138 35186	. 2 ⊢ (𝐷 ∈ 𝐴 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))
3	2	biimpi 219	1 ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))

This proof depends on syntax axioms:   → wi 4   ∨ wo 860   = wceq 1569   ∈ wcel 2142   ∪ cun 3902

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1824  ax-4 1838  ax-5 1939  ax-6 1996  ax-7 2037  ax-8 2144  ax-9 2152  ax-ext 2734

This proof depends on definitions:  df-bi 210  df-an 401  df-or 861  df-tru 1572  df-ex 1809  df-sb 2096  df-clab 2741  df-cleq 2754  df-clel 2837  df-v 3456  df-un 3909

This theorem is used by:  bnj1423  35448  bnj1452  35449