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Mirrors > Home > MPE Home > Th. List > ex-un | Structured version Visualization version GIF version |
Description: Example for df-un 3920. 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 4131 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ ({1} ∪ {8})) | |
2 | snsspr1 4779 | . . . . 5 ⊢ {1} ⊆ {1, 3} | |
3 | ssequn2 4148 | . . . . 5 ⊢ ({1} ⊆ {1, 3} ↔ ({1, 3} ∪ {1}) = {1, 3}) | |
4 | 2, 3 | mpbi 229 | . . . 4 ⊢ ({1, 3} ∪ {1}) = {1, 3} |
5 | 4 | uneq1i 4124 | . . 3 ⊢ (({1, 3} ∪ {1}) ∪ {8}) = ({1, 3} ∪ {8}) |
6 | 1, 5 | eqtr3i 2767 | . 2 ⊢ ({1, 3} ∪ ({1} ∪ {8})) = ({1, 3} ∪ {8}) |
7 | df-pr 4594 | . . 3 ⊢ {1, 8} = ({1} ∪ {8}) | |
8 | 7 | uneq2i 4125 | . 2 ⊢ ({1, 3} ∪ {1, 8}) = ({1, 3} ∪ ({1} ∪ {8})) |
9 | df-tp 4596 | . 2 ⊢ {1, 3, 8} = ({1, 3} ∪ {8}) | |
10 | 6, 8, 9 | 3eqtr4i 2775 | 1 ⊢ ({1, 3} ∪ {1, 8}) = {1, 3, 8} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1542 ∪ cun 3913 ⊆ wss 3915 {csn 4591 {cpr 4593 {ctp 4595 1c1 11059 3c3 12216 8c8 12221 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2708 |
This theorem depends on definitions: df-bi 206 df-an 398 df-or 847 df-tru 1545 df-ex 1783 df-sb 2069 df-clab 2715 df-cleq 2729 df-clel 2815 df-v 3450 df-un 3920 df-in 3922 df-ss 3932 df-pr 4594 df-tp 4596 |
This theorem is referenced by: ex-uni 29412 |
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