Theorem dfsn2 3748

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 3743	. 2 ⊢ {A, A} = ({A} ∪ {A})
2		unidm 3408	. 2 ⊢ ({A} ∪ {A}) = {A}
3	1, 2	eqtr2i 2374	1 ⊢ {A} = {A, A}

Syntax hints:   = wceq 1642   ∪ cun 3208  {csn 3738  {cpr 3739

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1546  ax-5 1557  ax-17 1616  ax-9 1654  ax-8 1675  ax-6 1729  ax-7 1734  ax-11 1746  ax-12 1925  ax-ext 2334

This theorem depends on definitions:  df-bi 177  df-or 359  df-an 360  df-nan 1288  df-tru 1319  df-ex 1542  df-nf 1545  df-sb 1649  df-clab 2340  df-cleq 2346  df-clel 2349  df-nfc 2479  df-v 2862  df-nin 3212  df-compl 3213  df-un 3215  df-pr 3743

This theorem is referenced by:  nfsn  3785  tpidm12  3822  tpidm  3825  unisn  3908  intsng  3962