Theorem opid 3851

Description: The ordered pair <. A , A >. in Kuratowski's representation. (Contributed by FL, 28-Dec-2011.)

Hypothesis

Ref	Expression
opid.1	|- A e. _V

Assertion

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

Proof of Theorem opid

Step	Hyp	Ref	Expression
1		dfsn2 3657	. . . 4 |- { A } = { A , A }
2	1	eqcomi 2211	. . 3 |- { A , A } = { A }
3	2	preq2i 3724	. 2 |- { { A } , { A , A } } = { { A } , { A } }
4		opid.1	. . 3 |- A e. _V
5	4, 4	dfop 3832	. 2 |- <. A , A >. = { { A } , { A , A } }
6		dfsn2 3657	. 2 |- { { A } } = { { A } , { A } }
7	3, 5, 6	3eqtr4i 2238	1 |- <. A , A >. = { { A } }

Syntax hints:   = wceq 1373   e. wcel 2178   _Vcvv 2776   {csn 3643   {cpr 3644   <.cop 3646

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 711  ax-5 1471  ax-7 1472  ax-gen 1473  ax-ie1 1517  ax-ie2 1518  ax-8 1528  ax-10 1529  ax-11 1530  ax-i12 1531  ax-bndl 1533  ax-4 1534  ax-17 1550  ax-i9 1554  ax-ial 1558  ax-i5r 1559  ax-ext 2189

This theorem depends on definitions:  df-bi 117  df-3an 983  df-tru 1376  df-nf 1485  df-sb 1787  df-clab 2194  df-cleq 2200  df-clel 2203  df-nfc 2339  df-v 2778  df-un 3178  df-sn 3649  df-pr 3650  df-op 3652

This theorem is referenced by:  dmsnsnsng  5179  funopg  5324  funopsn  5785