Theorem prcom 2447

Description: Commutative law for unordered pairs.

Assertion

Ref	Expression
prcom	|- {A, B} = {B, A}

Proof of Theorem prcom

Step	Hyp	Ref	Expression
1		uncom 2176	. 2 |- ({A} u. {B}) = ({B} u. {A})
2		df-pr 2413	. 2 |- {A, B} = ({A} u. {B})
3		df-pr 2413	. 2 |- {B, A} = ({B} u. {A})
4	1, 2, 3	3eqtr4 1505	1 |- {A, B} = {B, A}

Syntax hints:   = wceq 956   u. cun 2045  {csn 2409  {cpr 2410

This theorem is referenced by:  preq2 2449  pri2 2451  prprc2 2453  sspr 2475  preqr2 2482  preq12b 2483  opprc2 2499

This theorem was proved from axioms:  ax-1 4  ax-2 5  ax-3 6  ax-mp 7  ax-7 962  ax-gen 963  ax-8 964  ax-10 966  ax-12 968  ax-17 971  ax-4 973  ax-5o 975  ax-6o 978  ax-9o 1123  ax-10o 1140  ax-16 1210  ax-11o 1218  ax-ext 1459

This theorem depends on definitions:  df-bi 147  df-or 224  df-an 225  df-ex 981  df-sb 1172  df-clab 1464  df-cleq 1469  df-clel 1472  df-v 1812  df-un 2050  df-pr 2413

Copyright terms: Public domain