Theorem prcom 3420

Description: Commutative law for unordered pairs. (Contributed by NM, 5-Aug-1993.)

Assertion

Ref	Expression
prcom	⊢ {A, B} = {B, A}

Proof of Theorem prcom

Step	Hyp	Ref	Expression
1		uncom 3064	. 2 ⊢ ({A} ∪ {B}) = ({B} ∪ {A})
2		df-pr 3357	. 2 ⊢ {A, B} = ({A} ∪ {B})
3		df-pr 3357	. 2 ⊢ {B, A} = ({B} ∪ {A})
4	1, 2, 3	3eqtr4i 2052	1 ⊢ {A, B} = {B, A}

Syntax hints:   = wceq 1228   ∪ cun 2892  {csn 3350  {cpr 3351

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 617  ax-5 1316  ax-7 1317  ax-gen 1318  ax-ie1 1363  ax-ie2 1364  ax-8 1376  ax-10 1377  ax-11 1378  ax-i12 1379  ax-bnd 1380  ax-4 1381  ax-17 1400  ax-i9 1404  ax-ial 1409  ax-i5r 1410  ax-ext 2004

This theorem depends on definitions:  df-bi 110  df-tru 1231  df-nf 1330  df-sb 1628  df-clab 2009  df-cleq 2015  df-clel 2018  df-nfc 2149  df-v 2537  df-un 2899  df-pr 3357

This theorem is referenced by:  preq2  3422  tpcoma  3438  tpidm23  3445  prid2g  3449  prid2  3451  prprc2  3453  difprsn2  3478  preqr2g  3512  preqr2  3514  preq12b  3515  fvpr2  5291  fvpr2g  5293