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Mirrors > Home > MPE Home > Th. List > ex-un | Structured version Visualization version GIF version |
Description: Example for df-un 3940. 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 4141 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8})) | |
2 | snsspr1 4746 | . . . . 5 ⊢ {1} ⊆ {1, 3} | |
3 | ssequn2 4158 | . . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3}) | |
4 | 2, 3 | mpbi 232 | . . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3} |
5 | 4 | uneq1i 4134 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8}) |
6 | 1, 5 | eqtr3i 2846 | . 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8}) |
7 | df-pr 4569 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
8 | 7 | uneq2i 4135 | . 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8})) |
9 | df-tp 4571 | . 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8}) | |
10 | 6, 8, 9 | 3eqtr4i 2854 | 1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1533 ∪ cun 3933 ⊆ wss 3935 {csn 4566 {cpr 4568 {ctp 4570 1c1 10537 3c3 11692 8c8 11697 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1792 ax-4 1806 ax-5 1907 ax-6 1966 ax-7 2011 ax-8 2112 ax-9 2120 ax-10 2141 ax-11 2157 ax-12 2173 ax-ext 2793 |
This theorem depends on definitions: df-bi 209 df-an 399 df-or 844 df-tru 1536 df-ex 1777 df-nf 1781 df-sb 2066 df-clab 2800 df-cleq 2814 df-clel 2893 df-nfc 2963 df-v 3496 df-un 3940 df-in 3942 df-ss 3951 df-pr 4569 df-tp 4571 |
This theorem is referenced by: ex-uni 28204 |
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