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Mirrors > Home > ILE Home > Th. List > prcom | GIF version |
Description: Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.) |
Ref | Expression |
---|---|
prcom | ⊢ {A, B} = {B, A} |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | uncom 3081 | . 2 ⊢ ({A} ∪ {B}) = ({B} ∪ {A}) | |
2 | df-pr 3374 | . 2 ⊢ {A, B} = ({A} ∪ {B}) | |
3 | df-pr 3374 | . 2 ⊢ {B, A} = ({B} ∪ {A}) | |
4 | 1, 2, 3 | 3eqtr4i 2067 | 1 ⊢ {A, B} = {B, A} |
Colors of variables: wff set class |
Syntax hints: = wceq 1242 ∪ cun 2909 {csn 3367 {cpr 3368 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-mp 7 ax-ia1 99 ax-ia2 100 ax-ia3 101 ax-io 629 ax-5 1333 ax-7 1334 ax-gen 1335 ax-ie1 1379 ax-ie2 1380 ax-8 1392 ax-10 1393 ax-11 1394 ax-i12 1395 ax-bndl 1396 ax-4 1397 ax-17 1416 ax-i9 1420 ax-ial 1424 ax-i5r 1425 ax-ext 2019 |
This theorem depends on definitions: df-bi 110 df-tru 1245 df-nf 1347 df-sb 1643 df-clab 2024 df-cleq 2030 df-clel 2033 df-nfc 2164 df-v 2553 df-un 2916 df-pr 3374 |
This theorem is referenced by: preq2 3439 tpcoma 3455 tpidm23 3462 prid2g 3466 prid2 3468 prprc2 3470 difprsn2 3495 preqr2g 3529 preqr2 3531 preq12b 3532 fvpr2 5309 fvpr2g 5311 |
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