Theorem bnj1424 34821

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 34771	. 2 ⊢ (𝐷 ∈ 𝐴 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))
3	2	biimpi 216	1 ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))

Syntax hints:   → wi 4   ∨ wo 847   = wceq 1540   ∈ wcel 2109   ∪ cun 3909

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1795  ax-4 1809  ax-5 1910  ax-6 1967  ax-7 2008  ax-8 2111  ax-9 2119  ax-ext 2701

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 848  df-tru 1543  df-ex 1780  df-sb 2066  df-clab 2708  df-cleq 2721  df-clel 2803  df-v 3446  df-un 3916

This theorem is referenced by:  bnj1423  35034  bnj1452  35035