![]() |
Metamath Proof Explorer |
< Previous
Next >
Nearby theorems |
|
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 4165 | . 2 ⊢ (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) | |
2 | df-pr 4630 | . . 3 ⊢ {𝐴, 𝐵} = ({𝐴} ∪ {𝐵}) | |
3 | 2 | uneq1i 4158 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = (({𝐴} ∪ {𝐵}) ∪ {𝐶, 𝐷}) |
4 | tpass 4755 | . . 3 ⊢ {𝐵, 𝐶, 𝐷} = ({𝐵} ∪ {𝐶, 𝐷}) | |
5 | 4 | uneq2i 4159 | . 2 ⊢ ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) = ({𝐴} ∪ ({𝐵} ∪ {𝐶, 𝐷})) |
6 | 1, 3, 5 | 3eqtr4i 2768 | 1 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴} ∪ {𝐵, 𝐶, 𝐷}) |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1539 ∪ cun 3945 {csn 4627 {cpr 4629 {ctp 4631 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1795 ax-4 1809 ax-5 1911 ax-6 1969 ax-7 2009 ax-8 2106 ax-9 2114 ax-ext 2701 |
This theorem depends on definitions: df-bi 206 df-an 395 df-or 844 df-3or 1086 df-tru 1542 df-ex 1780 df-sb 2066 df-clab 2708 df-cleq 2722 df-clel 2808 df-v 3474 df-un 3952 df-sn 4628 df-pr 4630 df-tp 4632 |
This theorem is referenced by: en4 9285 ex-pw 29949 |
Copyright terms: Public domain | W3C validator |