Theorem dfsn2 3646

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 3639	. 2 ⊢ {𝐴, 𝐴} = ({𝐴} ∪ {𝐴})
2		unidm 3315	. 2 ⊢ ({𝐴} ∪ {𝐴}) = {𝐴}
3	1, 2	eqtr2i 2226	1 ⊢ {𝐴} = {𝐴, 𝐴}

Syntax hints:   = wceq 1372   ∪ cun 3163  {csn 3632  {cpr 3633

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 1469  ax-7 1470  ax-gen 1471  ax-ie1 1515  ax-ie2 1516  ax-8 1526  ax-10 1527  ax-11 1528  ax-i12 1529  ax-bndl 1531  ax-4 1532  ax-17 1548  ax-i9 1552  ax-ial 1556  ax-i5r 1557  ax-ext 2186

This theorem depends on definitions:  df-bi 117  df-tru 1375  df-nf 1483  df-sb 1785  df-clab 2191  df-cleq 2197  df-clel 2200  df-nfc 2336  df-v 2773  df-un 3169  df-pr 3639

This theorem is referenced by:  nfsn  3692  tpidm12  3731  tpidm  3734  preqsn  3815  opid  3836  unisn  3865  intsng  3918  opeqsn  4295  relop  4826  funopg  5302  enpr1g  6875  prfidceq  7007  hashprg  10934  bj-snexg  15712