Theorem prcom 3437

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 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}

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-bnd 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