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Mirrors > Home > NFE 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 3408 | . 2 ⊢ ({A} ∪ {B}) = ({B} ∪ {A}) | |
2 | df-pr 3742 | . 2 ⊢ {A, B} = ({A} ∪ {B}) | |
3 | df-pr 3742 | . 2 ⊢ {B, A} = ({B} ∪ {A}) | |
4 | 1, 2, 3 | 3eqtr4i 2383 | 1 ⊢ {A, B} = {B, A} |
Colors of variables: wff setvar class |
Syntax hints: = wceq 1642 ∪ cun 3207 {csn 3737 {cpr 3738 |
This theorem was proved from axioms: ax-1 5 ax-2 6 ax-3 7 ax-mp 8 ax-gen 1546 ax-5 1557 ax-17 1616 ax-9 1654 ax-8 1675 ax-6 1729 ax-7 1734 ax-11 1746 ax-12 1925 ax-ext 2334 |
This theorem depends on definitions: df-bi 177 df-or 359 df-an 360 df-nan 1288 df-tru 1319 df-ex 1542 df-nf 1545 df-sb 1649 df-clab 2340 df-cleq 2346 df-clel 2349 df-nfc 2478 df-v 2861 df-nin 3211 df-compl 3212 df-un 3214 df-pr 3742 |
This theorem is referenced by: preq2 3800 tpcoma 3816 tpidm23 3823 prid2g 3826 prid2 3828 difprsn2 3848 snprss2 4121 prprc1 4123 preqr2 4125 preq12b 4127 fvpr2 5450 |
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