Theorem dfsn2 2420

Description: Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15.

Assertion

Ref	Expression
dfsn2	|- {A} = {A, A}

Proof of Theorem dfsn2

Step	Hyp	Ref	Expression
1		df-pr 2413	. 2 |- {A, A} = ({A} u. {A})
2		unidm 2175	. 2 |- ({A} u. {A}) = {A}
3	1, 2	eqtr2 1496	1 |- {A} = {A, A}

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

This theorem is referenced by:  elsncg 2430  hbsn 2438  r19.12sn 2444  preqsn 2486  opprc2 2499  unisn 2517  intsn 2564  opprc3 2797  opeqsn 2802  relop 3275  dmsnsnsn 3329  funopg 3547  supsn 4591  boe 10460

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