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| Description: Example for df-un 3955. Example by David A. Wheeler. (Contributed by Mario Carneiro, 6-May-2015.) | 
| Ref | Expression | 
|---|---|
| ex-un | ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8} | 
| Step | Hyp | Ref | Expression | 
|---|---|---|---|
| 1 | unass 4171 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8})) | |
| 2 | snsspr1 4813 | . . . . 5 ⊢ {1} ⊆ {1, 3} | |
| 3 | ssequn2 4188 | . . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3}) | |
| 4 | 2, 3 | mpbi 230 | . . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3} | 
| 5 | 4 | uneq1i 4163 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8}) | 
| 6 | 1, 5 | eqtr3i 2766 | . 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8}) | 
| 7 | df-pr 4628 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
| 8 | 7 | uneq2i 4164 | . 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8})) | 
| 9 | df-tp 4630 | . 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8}) | |
| 10 | 6, 8, 9 | 3eqtr4i 2774 | 1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8} | 
| Colors of variables: wff setvar class | 
| Syntax hints: = wceq 1539 ∪ cun 3948 ⊆ wss 3950 {csn 4625 {cpr 4627 {ctp 4629 1c1 11157 3c3 12323 8c8 12328 | 
| 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 df-ss 3967 df-pr 4628 df-tp 4630 | 
| This theorem is referenced by: ex-uni 30446 | 
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