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Mirrors > Home > MPE Home > Th. List > ex-un | Structured version Visualization version GIF version |
Description: Example for df-un 3720. 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 3913 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8})) | |
2 | snsspr1 4490 | . . . . 5 ⊢ {1} ⊆ {1, 3} | |
3 | ssequn2 3929 | . . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3}) | |
4 | 2, 3 | mpbi 220 | . . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3} |
5 | 4 | uneq1i 3906 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8}) |
6 | 1, 5 | eqtr3i 2784 | . 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8}) |
7 | df-pr 4324 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
8 | 7 | uneq2i 3907 | . 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8})) |
9 | df-tp 4326 | . 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8}) | |
10 | 6, 8, 9 | 3eqtr4i 2792 | 1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1632 ∪ cun 3713 ⊆ wss 3715 {csn 4321 {cpr 4323 {ctp 4325 1c1 10149 3c3 11283 8c8 11288 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1871 ax-4 1886 ax-5 1988 ax-6 2054 ax-7 2090 ax-9 2148 ax-10 2168 ax-11 2183 ax-12 2196 ax-13 2391 ax-ext 2740 |
This theorem depends on definitions: df-bi 197 df-or 384 df-an 385 df-tru 1635 df-ex 1854 df-nf 1859 df-sb 2047 df-clab 2747 df-cleq 2753 df-clel 2756 df-nfc 2891 df-v 3342 df-un 3720 df-in 3722 df-ss 3729 df-pr 4324 df-tp 4326 |
This theorem is referenced by: ex-uni 27615 |
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