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Mirrors > Home > ILE Home > Th. List > qdass | GIF version |
Description: Two ways to write an unordered quadruple. (Contributed by Mario Carneiro, 5-Jan-2016.) |
Ref | Expression |
---|---|
qdass | ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷}) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | unass 3228 | . 2 ⊢ (({𝐴, 𝐵} ∪ {𝐶}) ∪ {𝐷}) = ({𝐴, 𝐵} ∪ ({𝐶} ∪ {𝐷})) | |
2 | df-tp 3530 | . . 3 ⊢ {𝐴, 𝐵, 𝐶} = ({𝐴, 𝐵} ∪ {𝐶}) | |
3 | 2 | uneq1i 3221 | . 2 ⊢ ({𝐴, 𝐵, 𝐶} ∪ {𝐷}) = (({𝐴, 𝐵} ∪ {𝐶}) ∪ {𝐷}) |
4 | df-pr 3529 | . . 3 ⊢ {𝐶, 𝐷} = ({𝐶} ∪ {𝐷}) | |
5 | 4 | uneq2i 3222 | . 2 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵} ∪ ({𝐶} ∪ {𝐷})) |
6 | 1, 3, 5 | 3eqtr4ri 2169 | 1 ⊢ ({𝐴, 𝐵} ∪ {𝐶, 𝐷}) = ({𝐴, 𝐵, 𝐶} ∪ {𝐷}) |
Colors of variables: wff set class |
Syntax hints: = wceq 1331 ∪ cun 3064 {csn 3522 {cpr 3523 {ctp 3524 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-ia1 105 ax-ia2 106 ax-ia3 107 ax-io 698 ax-5 1423 ax-7 1424 ax-gen 1425 ax-ie1 1469 ax-ie2 1470 ax-8 1482 ax-10 1483 ax-11 1484 ax-i12 1485 ax-bndl 1486 ax-4 1487 ax-17 1506 ax-i9 1510 ax-ial 1514 ax-i5r 1515 ax-ext 2119 |
This theorem depends on definitions: df-bi 116 df-tru 1334 df-nf 1437 df-sb 1736 df-clab 2124 df-cleq 2130 df-clel 2133 df-nfc 2268 df-v 2683 df-un 3070 df-pr 3529 df-tp 3530 |
This theorem is referenced by: (None) |
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