Theorem dfsn2 3636

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 3629	. 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴})
2		unidm 3306	. 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴}
3	1, 2	eqtr2i 2218	1 ⊢ {𝐴} = {𝐴, 𝐴}

Syntax hints:   = wceq 1364   ∪ cun 3155  {csn 3622  {cpr 3623

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-io 710  ax-5 1461  ax-7 1462  ax-gen 1463  ax-ie1 1507  ax-ie2 1508  ax-8 1518  ax-10 1519  ax-11 1520  ax-i12 1521  ax-bndl 1523  ax-4 1524  ax-17 1540  ax-i9 1544  ax-ial 1548  ax-i5r 1549  ax-ext 2178

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1475  df-sb 1777  df-clab 2183  df-cleq 2189  df-clel 2192  df-nfc 2328  df-v 2765  df-un 3161  df-pr 3629

This theorem is referenced by:  nfsn  3682  tpidm12  3721  tpidm  3724  preqsn  3805  opid  3826  unisn  3855  intsng  3908  opeqsn  4285  relop  4816  funopg  5292  enpr1g  6857  prfidceq  6989  hashprg  10900  bj-snexg  15558