Theorem dfop 3764

Description: Value of an ordered pair when the arguments are sets, with the conclusion corresponding to Kuratowski's original definition. (Contributed by NM, 25-Jun-1998.)

Hypotheses

Ref	Expression
dfop.1	|- A e. _V
dfop.2	|- B e. _V

Assertion

Ref	Expression
dfop	|- <. A , B >. = { { A } , { A , B } }

Proof of Theorem dfop

Step	Hyp	Ref	Expression
1		dfop.1	. 2 |- A e. _V
2		dfop.2	. 2 |- B e. _V
3		dfopg 3763	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. = { { A } , { A , B } } )
4	1, 2, 3	mp2an 424	1 |- <. A , B >. = { { A } , { A , B } }

Syntax hints:   = wceq 1348   e. wcel 2141   _Vcvv 2730   {csn 3583   {cpr 3584   <.cop 3586

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 105  ax-ia2 106  ax-ia3 107  ax-5 1440  ax-7 1441  ax-gen 1442  ax-ie1 1486  ax-ie2 1487  ax-8 1497  ax-11 1499  ax-4 1503  ax-17 1519  ax-i9 1523  ax-ial 1527  ax-i5r 1528  ax-ext 2152

This theorem depends on definitions:  df-bi 116  df-3an 975  df-tru 1351  df-nf 1454  df-sb 1756  df-clab 2157  df-cleq 2163  df-clel 2166  df-v 2732  df-op 3592

This theorem is referenced by:  opid  3783  elop  4216  opi1  4217  opi2  4218  opeqsn  4237  opeqpr  4238  uniop  4240  op1stb  4463  xpsspw  4723  relop  4761  funopg  5232