Theorem dfsn2 4642

Description: Alternate definition of singleton. Definition 5.1 of [TakeutiZaring] p. 15. (Contributed by NM, 24-Apr-1994.)

Assertion

Ref	Expression
dfsn2	⊢ {𝐴} = {𝐴, 𝐴}

Proof of Theorem dfsn2

Step	Hyp	Ref	Expression
1		df-pr 4632	. 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴})
2		unidm 4153	. 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴}
3	1, 2	eqtr2i 2762	1 ⊢ {𝐴} = {𝐴, 𝐴}

Syntax hints:   = wceq 1542   ∪ cun 3947  {csn 4629  {cpr 4631

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1798  ax-4 1812  ax-5 1914  ax-6 1972  ax-7 2012  ax-8 2109  ax-9 2117  ax-ext 2704

This theorem depends on definitions:  df-bi 206  df-an 398  df-or 847  df-tru 1545  df-ex 1783  df-sb 2069  df-clab 2711  df-cleq 2725  df-clel 2811  df-v 3477  df-un 3954  df-pr 4632

This theorem is referenced by:  nfsn  4712  disjprsn  4719  tpidm12  4760  tpidm  4763  ifpprsnss  4769  preqsnd  4860  elpreqprlem  4867  opidg  4893  unisng  4930  intsng  4990  vsnex  5430  opeqsng  5504  propeqop  5508  relop  5851  funopg  6583  f1oprswap  6878  fnprb  7210  enpr1g  9020  supsn  9467  infsn  9500  pr2ne  9999  prdom2  10001  wuntp  10706  wunsn  10711  grusn  10799  prunioo  13458  hashprg  14355  hashfun  14397  hashle2pr  14438  lcmfsn  16572  lubsn  18435  indislem  22503  hmphindis  23301  wilthlem2  26573  upgrex  28352  umgrnloop0  28369  edglnl  28403  usgrnloop0ALT  28462  uspgr1v1eop  28506  1loopgruspgr  28757  1egrvtxdg0  28768  umgr2v2eedg  28781  umgr2v2e  28782  ifpsnprss  28880  upgriswlk  28898  clwwlkn1  29294  upgr1wlkdlem1  29398  1to2vfriswmgr  29532  esumpr2  33065  dvh2dim  40316  wopprc  41769  clsk1indlem4  42795  sge0prle  45117  meadjun  45178  elsprel  46143  upgrwlkupwlk  46518