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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 3770 | . 2 ⊢ (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) | |
2 | df-pr 4180 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 2 | uneq1i 3763 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) |
4 | tpass 4287 | . . 3 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵} ∪ {𝐶, 𝐷}) | |
5 | 4 | uneq2i 3764 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) |
6 | 1, 3, 5 | 3eqtr4i 2654 | 1 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1483 ∪ cun 3572 {csn 4177 {cpr 4179 {ctp 4181 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1722 ax-4 1737 ax-5 1839 ax-6 1888 ax-7 1935 ax-9 1999 ax-10 2019 ax-11 2034 ax-12 2047 ax-13 2246 ax-ext 2602 |
This theorem depends on definitions: df-bi 197 df-or 385 df-an 386 df-3or 1038 df-tru 1486 df-ex 1705 df-nf 1710 df-sb 1881 df-clab 2609 df-cleq 2615 df-clel 2618 df-nfc 2753 df-v 3202 df-un 3579 df-sn 4178 df-pr 4180 df-tp 4182 |
This theorem is referenced by: en4 8198 ex-pw 27286 |
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