Theorem dfop 3903

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 3902	. 2 |- ( ( A e. _V /\ B e. _V ) -> <. A , B >. = { { A } , { A , B } } )
4	1, 2, 3	mp2an 430	1 |- <. A , B >. = { { A } , { A , B } }

This proof depends on syntax axioms:   = wceq 1402   e. wcel 2209   _Vcvv 2821   {csn 3709   {cpr 3710   <.cop 3712

This proof depends on axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-ia1 106  ax-ia2 107  ax-ia3 108  ax-5 1500  ax-7 1501  ax-gen 1502  ax-ie1 1546  ax-ie2 1547  ax-8 1557  ax-11 1559  ax-4 1563  ax-17 1579  ax-i9 1583  ax-ial 1587  ax-i5r 1588  ax-ext 2220

This proof depends on definitions:  df-bi 117  df-3an 1011  df-tru 1405  df-nf 1514  df-sb 1816  df-clab 2225  df-cleq 2231  df-clel 2234  df-v 2823  df-op 3718

This theorem is used by:  opid  3922  elop  4371  opi1  4372  opi2  4373  opeqsn  4393  opeqpr  4394  uniop  4396  op1stb  4624  xpsspw  4887  relop  4930  funopg  5411  funopsn  5891