Theorem dfop 3818

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 3817	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. = { { A } , { A , B } } )
4	1, 2, 3	mp2an 426	1 |- <. A , B >. = { { A } , { A , B } }

Syntax hints:   = wceq 1373   e. wcel 2176   _Vcvv 2772   {csn 3633   {cpr 3634   <.cop 3636

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-5 1470  ax-7 1471  ax-gen 1472  ax-ie1 1516  ax-ie2 1517  ax-8 1527  ax-11 1529  ax-4 1533  ax-17 1549  ax-i9 1553  ax-ial 1557  ax-i5r 1558  ax-ext 2187

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1484  df-sb 1786  df-clab 2192  df-cleq 2198  df-clel 2201  df-v 2774  df-op 3642

This theorem is referenced by:  opid  3837  elop  4276  opi1  4277  opi2  4278  opeqsn  4298  opeqpr  4299  uniop  4301  op1stb  4526  xpsspw  4788  relop  4829  funopg  5306  funopsn  5764