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Mirrors > Home > MPE Home > Th. List > ex-un | Structured version Visualization version GIF version |
Description: Example for df-un 3836. 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 4033 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8})) | |
2 | snsspr1 4622 | . . . . 5 ⊢ {1} ⊆ {1, 3} | |
3 | ssequn2 4049 | . . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3}) | |
4 | 2, 3 | mpbi 222 | . . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3} |
5 | 4 | uneq1i 4026 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8}) |
6 | 1, 5 | eqtr3i 2804 | . 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8}) |
7 | df-pr 4445 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
8 | 7 | uneq2i 4027 | . 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8})) |
9 | df-tp 4447 | . 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8}) | |
10 | 6, 8, 9 | 3eqtr4i 2812 | 1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1507 ∪ cun 3829 ⊆ wss 3831 {csn 4442 {cpr 4444 {ctp 4446 1c1 10338 3c3 11499 8c8 11504 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1758 ax-4 1772 ax-5 1869 ax-6 1928 ax-7 1965 ax-8 2052 ax-9 2059 ax-10 2079 ax-11 2093 ax-12 2106 ax-ext 2750 |
This theorem depends on definitions: df-bi 199 df-an 388 df-or 834 df-tru 1510 df-ex 1743 df-nf 1747 df-sb 2016 df-clab 2759 df-cleq 2771 df-clel 2846 df-nfc 2918 df-v 3417 df-un 3836 df-in 3838 df-ss 3845 df-pr 4445 df-tp 4447 |
This theorem is referenced by: ex-uni 27986 |
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