Theorem bnj1138 33131

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 2829	. 2 ⊢ (𝑋 ∈ 𝐴 ↔ 𝑋 ∈ (𝐵 ∪ 𝐶))
3		elun 4103	. 2 ⊢ (𝑋 ∈ (𝐵 ∪ 𝐶) ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))
4	2, 3	bitri 275	1 ⊢ (𝑋 ∈ 𝐴 ↔ (𝑋 ∈ 𝐵 ∨ 𝑋 ∈ 𝐶))

Syntax hints:   ↔ wb 205   ∨ wo 845   = wceq 1541   ∈ wcel 2106   ∪ cun 3903

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 1913  ax-6 1971  ax-7 2011  ax-8 2108  ax-9 2116  ax-ext 2708

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 846  df-tru 1544  df-ex 1782  df-sb 2068  df-clab 2715  df-cleq 2729  df-clel 2815  df-v 3445  df-un 3910

This theorem is referenced by:  bnj1424  33181  bnj1408  33379  bnj1417  33384