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Mirrors > Home > MPE Home > Th. List > qdassr | Structured version Visualization version GIF version |
Description: Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
qdassr | ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unass 4162 | . 2 ⊢ (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) | |
2 | df-pr 4625 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 2 | uneq1i 4155 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) |
4 | tpass 4749 | . . 3 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵} ∪ {𝐶, 𝐷}) | |
5 | 4 | uneq2i 4156 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) |
6 | 1, 3, 5 | 3eqtr4i 2769 | 1 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1541 ∪ cun 3942 {csn 4622 {cpr 4624 {ctp 4626 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1913 ax-6 1971 ax-7 2011 ax-8 2108 ax-9 2116 ax-ext 2702 |
This theorem depends on definitions: df-bi 206 df-an 397 df-or 846 df-3or 1088 df-tru 1544 df-ex 1782 df-sb 2068 df-clab 2709 df-cleq 2723 df-clel 2809 df-v 3475 df-un 3949 df-sn 4623 df-pr 4625 df-tp 4627 |
This theorem is referenced by: en4 9266 ex-pw 29547 |
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