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| Description: Inference from membership to union. (Contributed by NM, 21-Jun-1993.) | 
| Ref | Expression | 
|---|---|
| uneqri.1 | ⊢ ((𝑥 ∈ 𝐴 ∨ 𝑥 ∈ 𝐵) ↔ 𝑥 ∈ 𝐶) | 
| Ref | Expression | 
|---|---|
| uneqri | ⊢ (𝐴 ∪ 𝐵) = 𝐶 | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | elun 4152 | . . 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 2733 | 1 ⊢ (𝐴 ∪ 𝐵) = 𝐶 | 
| Colors of variables: wff setvar class | 
| Syntax hints: ↔ wb 206 ∨ wo 847 = wceq 1539 ∈ wcel 2107 ∪ cun 3948 | 
| This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1794 ax-4 1808 ax-5 1909 ax-6 1966 ax-7 2006 ax-8 2109 ax-9 2117 ax-ext 2707 | 
| This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-tru 1542 df-ex 1779 df-sb 2064 df-clab 2714 df-cleq 2728 df-clel 2815 df-v 3481 df-un 3955 | 
| This theorem is referenced by: unidm 4156 uncom 4157 unass 4171 dfun2 4269 undi 4284 unab 4307 un0 4393 inundif 4478 | 
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