Theorem bnj1424 30163

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 30113	. 2 ⊢ (𝐷 ∈ 𝐴 ↔ (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))
3	2	biimpi 205	1 ⊢ (𝐷 ∈ 𝐴 → (𝐷 ∈ 𝐵 ∨ 𝐷 ∈ 𝐶))

Syntax hints:   → wi 4   ∨ wo 382   = wceq 1475   ∈ wcel 1977   ∪ cun 3538

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1713  ax-4 1728  ax-5 1827  ax-6 1875  ax-7 1922  ax-10 2006  ax-11 2021  ax-12 2034  ax-13 2234  ax-ext 2590

This theorem depends on definitions:  df-bi 196  df-or 384  df-an 385  df-tru 1478  df-ex 1696  df-nf 1701  df-sb 1868  df-clab 2597  df-cleq 2603  df-clel 2606  df-nfc 2740  df-v 3175  df-un 3545

This theorem is referenced by:  bnj1423  30373  bnj1452  30374