|   | Metamath Proof Explorer | < Previous  
      Next > Nearby theorems | |
| Mirrors > Home > MPE Home > Th. List > un23 | Structured version Visualization version GIF version | ||
| Description: A rearrangement of union. (Contributed by NM, 12-Aug-2004.) (Proof shortened by Andrew Salmon, 26-Jun-2011.) | 
| Ref | Expression | 
|---|---|
| un23 | ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | unass 4171 | . 2 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = (𝐴 ∪ (𝐵 ∪ 𝐶)) | |
| 2 | un12 4172 | . 2 ⊢ (𝐴 ∪ (𝐵 ∪ 𝐶)) = (𝐵 ∪ (𝐴 ∪ 𝐶)) | |
| 3 | uncom 4157 | . 2 ⊢ (𝐵 ∪ (𝐴 ∪ 𝐶)) = ((𝐴 ∪ 𝐶) ∪ 𝐵) | |
| 4 | 1, 2, 3 | 3eqtri 2768 | 1 ⊢ ((𝐴 ∪ 𝐵) ∪ 𝐶) = ((𝐴 ∪ 𝐶) ∪ 𝐵) | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∪ 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: ssunpr 4833 setscom 17218 cycpmco2rn 33146 poimirlem6 37634 poimirlem7 37635 poimirlem16 37644 poimirlem19 37647 iocunico 43228 dfrcl2 43692 | 
| Copyright terms: Public domain | W3C validator |