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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 4182 | . 2 ⊢ (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) | |
2 | df-pr 4634 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 2 | uneq1i 4174 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) |
4 | tpass 4757 | . . 3 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵} ∪ {𝐶, 𝐷}) | |
5 | 4 | uneq2i 4175 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) |
6 | 1, 3, 5 | 3eqtr4i 2773 | 1 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1537 ∪ cun 3961 {csn 4631 {cpr 4633 {ctp 4635 |
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 1908 ax-6 1965 ax-7 2005 ax-8 2108 ax-9 2116 ax-ext 2706 |
This theorem depends on definitions: df-bi 207 df-an 396 df-or 848 df-3or 1087 df-tru 1540 df-ex 1777 df-sb 2063 df-clab 2713 df-cleq 2727 df-clel 2814 df-v 3480 df-un 3968 df-sn 4632 df-pr 4634 df-tp 4636 |
This theorem is referenced by: en4 9315 ex-pw 30458 |
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