Theorem uneqri 4107

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 4104	. . 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 2732	1 ⊢ (𝐴 ∪ 𝐵) = 𝐶

Syntax hints:   ↔ wb 206   ∨ wo 848   = wceq 1542   ∈ wcel 2114   ∪ cun 3898

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 1912  ax-6 1969  ax-7 2010  ax-8 2116  ax-9 2124  ax-ext 2707

This theorem depends on definitions:  df-bi 207  df-an 396  df-or 849  df-tru 1545  df-ex 1782  df-sb 2069  df-clab 2714  df-cleq 2727  df-clel 2810  df-v 3441  df-un 3905

This theorem is referenced by:  unidm  4108  uncom  4109  unass  4123  dfun2  4221  undi  4236  unab  4259  un0  4345  inundif  4430