Theorem uneqri 4151

Description: Inference from membership to union. (Contributed by NM, 21-Jun-1993.)

Hypothesis

Ref	Expression
uneqri.1	⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)

Assertion

Ref	Expression
uneqri	⊢ (𝐴 ∪ 𝐵) = 𝐶

Distinct variable groups:   𝑥,𝐴   𝑥,𝐵   𝑥,𝐶

Proof of Theorem uneqri

Step	Hyp	Ref	Expression
1		elun 4148	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		uneqri.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)
3	1, 2	bitri 275	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ 𝐶)
4	3	eqriv 2730	1 ⊢ (𝐴 ∪ 𝐵) = 𝐶

Syntax hints:   ↔ wb 205   ∨ wo 846   = wceq 1542   ∈ wcel 2107   ∪ cun 3946

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3953

This theorem is referenced by:  unidm  4152  uncom  4153  unass  4166  dfun2  4259  undi  4274  unab  4298  un0  4390  inundif  4478