Theorem uneqri 4158

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 4155	. . 3 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ (𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵))
2		uneqri.1	. . 3 ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶)
3	1, 2	bitri 278	. 2 ⊢ (𝑥 ∈ (𝐴 ∪ 𝐵) ↔ 𝑥 ∈ 𝐶)
4	3	eqriv 2760	1 ⊢ (𝐴 ∪ 𝐵) = 𝐶

This proof depends on syntax axioms:   ↔ wb 209   ∨ wo 861   = wceq 1570   ∈ wcel 2147   ∪ cun 4152

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1828  ax-4 1842  ax-5 1945  ax-6 2002  ax-7 2043  ax-8 2149  ax-9 2157  ax-ext 2735

This proof depends on definitions:  df-bi 210  df-an 402  df-or 862  df-tru 1573  df-ex 1813  df-sb 2102  df-clab 2742  df-cleq 2755  df-clel 2839  df-v 3456  df-un 4154

This theorem is used by:  unidm  4159  uncom  4160  unass  4173  dfun2  4218  undi  4233  unab  4256  un0  4347  inundif  4438