Theorem prcom 1840

Description: Commutative law for unordered pairs.

Assertion

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

Proof of Theorem prcom

Step	Hyp	Ref	Expression
1		uncom 1604	. 2 ⊢ ({A} ∪ {B}) = ({B} ∪ {A})
2		df-pr 1812	. 2 ⊢ {A, B} = ({A} ∪ {B})
3		df-pr 1812	. 2 ⊢ {B, A} = ({B} ∪ {A})
4	1, 2, 3	3eqtr4 1126	1 ⊢ {A, B} = {B, A}

Syntax hints:   = wceq 1091   ∪ cun 1485  {csn 1808  {cpr 1809

This theorem is referenced by:  pri2 1842  preq2 1871  prer2 1873  preq12b 1874  prex 1892  opprc2 1907

This theorem was proved from axioms:  ax-1 3  ax-2 4  ax-3 5  ax-mp 6  ax-4 673  ax-5 674  ax-6 675  ax-7 676  ax-gen 677  ax-8 798  ax-9 799  ax-10 800  ax-11 801  ax-12 802  ax-16 922  ax-17 925  ax-ext 1074

This theorem depends on definitions:  df-bi 128  df-or 197  df-an 198  df-ex 679  df-sb 853  df-clab 1093  df-cleq 1097  df-clel 1099  df-v 1349  df-un 1490  df-pr 1812

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