Theorem bnj1138 32170

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

Hypothesis

Ref	Expression
bnj1138.1	⊢ 𝐴 = (𝐵 ∪ 𝐶)

Assertion

Ref	Expression
bnj1138	⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))

Proof of Theorem bnj1138

Step	Hyp	Ref	Expression
1		bnj1138.1	. . 3 ⊢ 𝐴 = (𝐵 ∪ 𝐶)
2	1	eleq2i 2881	. 2 ⊢ (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (𝐵 ∪ 𝐶))
3		elun 4076	. 2 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))
4	2, 3	bitri 278	1 ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))

Syntax hints:   ↔ wb 209   ∨ wo 844   = wceq 1538   ∈ wcel 2111   ∪ cun 3879

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 1911  ax-6 1970  ax-7 2015  ax-8 2113  ax-9 2121  ax-ext 2770

This theorem depends on definitions:  df-bi 210  df-an 400  df-or 845  df-ex 1782  df-sb 2070  df-clab 2777  df-cleq 2791  df-clel 2870  df-v 3443  df-un 3886

This theorem is referenced by:  bnj1424  32220  bnj1408  32418  bnj1417  32423