Theorem dfsn2 3632

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 3625	. 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴})
2		unidm 3302	. 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴}
3	1, 2	eqtr2i 2215	1 ⊢ {𝐴} = {𝐴, 𝐴}

Syntax hints:   = wceq 1364   ∪ cun 3151  {csn 3618  {cpr 3619

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 1458  ax-7 1459  ax-gen 1460  ax-ie1 1504  ax-ie2 1505  ax-8 1515  ax-10 1516  ax-11 1517  ax-i12 1518  ax-bndl 1520  ax-4 1521  ax-17 1537  ax-i9 1541  ax-ial 1545  ax-i5r 1546  ax-ext 2175

This theorem depends on definitions:  df-bi 117  df-tru 1367  df-nf 1472  df-sb 1774  df-clab 2180  df-cleq 2186  df-clel 2189  df-nfc 2325  df-v 2762  df-un 3157  df-pr 3625

This theorem is referenced by:  nfsn  3678  tpidm12  3717  tpidm  3720  preqsn  3801  opid  3822  unisn  3851  intsng  3904  opeqsn  4281  relop  4812  funopg  5288  enpr1g  6852  hashprg  10879  bj-snexg  15404